2D arrays of diamond shaped cells having multiple josephson junctions

ABSTRACT

A two-dimensional SQIF array and methods for manufacture can include at least two bi-SQUIDs that share an inductance. The bi-SQUIDs can be combined to establish a diamond-shaped cell. A plurality of the diamond shaped cells can be packed tightly together so that each cell shares at least three cell junctions with adjacent cells to establish the SQIF array. Because of the close proximity of the cells, the effect that the mutual inductances each cell has on adjacent cells can be accounted for, as well as the SQIF array boundary conditions along the array edges. To do this, a matrix of differential equations can be solved to provide for the recommended inductance of each bi-SQUID in the SQIF array. Each bi-SQUID can be manufactured with the recommended inductance to result in a SQIF having an increased strength of anti-peak response, but without sacrificing the linearity of the response.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a continuation-in-part of U.S. application Ser. No.13/686,994, filed Nov. 28, 2012, and entitled “Linear Voltage Responseof Non-Uniform Arrays of Bi-SQUIDs”. The '994 application is herebyincorporated by reference herein in its entirety

FEDERALLY-SPONSORED RESEARCH AND DEVELOPMENT

This invention (Navy Case No. 102297) is assigned to the United StatesGovernment and is available for licensing for commercial purposes.Licensing and technical inquires may be directed to the Office ofResearch and Technical Applications, Space and Naval Warfare SystemsCenter, Pacific, Code 72120, San Diego, Calif. 92152; voice (619)553-5118; email ssc pac T2@navy.mil.

FIELD OF THE INVENTION

The present invention pertains generally to mechanisms and methods forimproving signal detection and amplification. More specifically, thepresent invention pertains to Superconducting Quantum InterferenceFilters (SQIFs). The invention is particularly, but not exclusively,useful as a SQIF having multiple Superconductive Quantum InterferenceDevices (SQUIDs), which can be arranged in a two-dimensional array sothat adjacent bi-SQUIDs share inductances.

BACKGROUND OF THE INVENTION

The SQUID is one of the most sensitive magnetic field devices in theprior art, and it can be used for wide range of applications, includingbiology, medicine, geology, systems for semiconductor circuitdiagnostics, security, magnetic resonance imaging (MRI) and evencosmology research. In recent years, arrays of coupled oscillators havebeen considered as a general mechanism for improving signal detectionand amplification. Theoretical and experimental studies can beinterpreted to show that arrays of SQUIDs can yield comparableimprovements in signal output relative to background noise, over thoseof a single SQUID device.

A peculiar configuration that has gained considerable attention amongthe physics and engineering community is that of multi-loop arrays ofJosephson Junctions (JJs) with non-uniformly distributed loop areas.Typically, each loop contains two JJs, i.e., a standard DC-SQUID, butthe loop size varies from loop to loop. These types of unconventionalgeometric structures of JJs are known to exhibit a magnetic fluxdependent voltage response V(φ_(e)), where φ_(e) denotes an externalmagnetic flux normalized by the quantum flux, that has a pronouncedsingle peak with a large voltage swing at zero magnetic field. Thepotential high dynamic range and linearity of the “anti-peak” voltageresponse render the array an ideal detector of absolute strength ofexternal magnetic fields. These arrays are also commonly known asSuperconducting Quantum Interference Filters (SQIFs).

Improving the linearity of SQIFs is critical for developing advancedtechnologies, such as low noise amplifiers (LNA's) that can furtherincrease link margins and affect an entire communication system. SQIFscan also be used in unmanned aerial vehicles (UAVs), where size, weightand power are limited, and as “electrically small” antennas to provideacceptable gain for the antenna. SQIFs can also be used in land minedetection applications. But for all of these applications, it is desiredto improve the linear response of the SQIF device.

The quest to increase the linearity of SQUID and SQIF arrays was boostedby the introduction of the bi-SQUID, which has three 3 JJ's. Anon-linear inductance of the additional third JJ can provide a desiredlinearizing effect for the orverall SQIF. These bi-SQUIDs are now beingused in uniform and non-uniform (SQIF) arrays in place of conventionaldc SQUIDs with a goal of achieving higher linearity. However, most ofthe design efforts to date that use SQUIDs and/or bi-SQUIDs are directedat the optimization of one-dimensional (1D) serial or parallel arraysand their combinations. This is due to the higher complexity analysisand modeling required for 2D arrays to account for mutual coupling thatoccurs between neighboring cells and complex current distribution inarrays. What is desired is a two-dimensnional array of bi-SQUIDs.

In view of the above, it is an object of the present invention toprovide a two-dimensional (2D) SQIF array having a tightly coupled 2Dnetwork of bi-SQUID cells, in which junctions and inductances are sharedbetween adjacent bi-SQUID cells. Another object of the present inventionis to provide a two-dimensional (2D) SQIF array having diamond shaped,dual bi-SQUID cells. Still another object of the present invention is toprovide a 2D SQIF that maximizes the number of bi-SQUID cells which canbe placed with a given SQIF array area. Yet another object of thepresent invention is to maximize the voltage response and dynamic rangeof a SQIF by manipulating the critical current, inductive couplingbetween loops, number of loops, bias current, and distribution of loopareas of the array cells of diamond bi-SQUIDs. Still another object ofthe present invention is to provide a 2D SQIF array with increased highdynamic range and linearity of the “anti-peak” response, for use inmultiple applications such as magnetic field detectors, magneticantennas and wide-band low-noise amplifiers. Another object of thepresent invention is to provide a SQIF and methods for manufacture thatcan be easily tailored in a cost-effective manner to result in a SQIFhaving bi-SQUID array cells that has been optimized according to theuser's needs.

SUMMARY OF THE INVENTION

A two-dimensional SQIF array and methods for manufacture according toseveral embodiments of the present invention can include at least twobi-SQUIDs that share an inductance. The two bi-SQUIDs can be combined toestablish a cell that is diamond shaped when viewed in top plan. Aplurality of the diamond shaped cells can be packed tightly together toestablish the SQIF array. With this configuration, each cell shares atleast three cell junctions with adjacent cells.

Because of the close proximity of the cells, the effect that the mutualinductances each cell has on adjacent cells can be accounted for.Additionally, the boundary conditions along the edges can be determined.To do this, a matrix of sixty-six 66 differential equations can besolved to provide for the recommended inductance of each bi-SQUID in the2D SQIF array. Each bi-SQUID can be manufactured with the recommendedinductance to result in a SQIF having an increased strength of anti-peakresponse, but without sacrificing the linearity of the response.

BRIEF DESCRIPTION OF THE DRAWINGS

The novel features of the present invention will be best understood fromthe accompanying drawings, taken in conjunction with the accompanyingdescription, in which similarly-referenced characters refer tosimilarly-referenced parts, and in which:

FIG. 1 is a schematic diagram of a direct current SuperconductiveQuantum Interference Devices (DC-SQUID) as known in the prior art;

FIG. 2 is a schematic diagram of a prior art bi-SQUID;

FIG. 3 is a graphical representation of the voltage response of theDC-SQUID and bi-SQUID of FIGS. 1 and 2, as a function of magnetic fieldstrength;

FIG. 4 is a schematic diagram of a single cell dual bi-SQUID of the 2DArray of the present invention according to several embodiments;

FIG. 5 is a graphical representation of the voltage response of thesingle cell dual bi-SQUID of FIG. 4 as a function of magnetic fieldstrength;

FIG. 6 is a top plan view of the single cell dual bi-SQUID of FIG. 4;

FIG. 7 is a schematic view of the 2D SQIF array of the present inventionaccording to several embodiments, which illustrate a plurality of thecells of FIGS. 4 and 5;

FIGS. 8a-8i are schematic views of the various boundary conditions forthe 2D SQIF array of FIG. 7;

FIG. 9 is a top plan view of a portion of the 2D SQIF array of FIG. 7;

FIG. 10 is a graphical representation of the voltage response of the 2DSQIF array of FIG. 7 as a function of magnetic field strength;

FIG. 11 is a graphical representation of graphical representation of theflux noise spectral density versus frequency at the mid-point of thepositive slope of the anti-peak in FIG. 10;

FIG. 12 is a graphical representation of energy sensitivity versusfrequency at the mid-point of the positive slope of the anti-peak inFIG. 10;

FIG. 13 is a graphical representation of noise temperature at themid-point of the positive slope of the anti-peak in FIG. 10; and,

FIG. 14 is a block diagram, which illustrates steps that can be taken toaccomplish the methods of the present invention according to severalembodiments.

DETAILED DESCRIPTION OF THE EMBODIMENTS

A. Prior Art.

In brief overview, and referring initially to FIGS. 1 and 2, a DC SQUID10 and DC bi-SQUID 20 are shown. FIG. 1 is a schematic diagram of asingle DC SQUID 10. DC SQUID 10 can have two Josephson Junctions (JJ's)12 a, 12 b arranged in parallel, connected with superconductingmaterial, as represented by the schematic diagram in FIG. 1. A DCbi-SQUID 20, which is a SQUID with an additional Josephson junction 12 cbisecting the superconducting loop, was introduced as an alternative totraditional SQUIDs. DC bi-SQUIDs have shown superior linearity in theaverage voltage response anti-peak feature. In FIG. 2, (i_(b), i₁, i₂,i₃, i₄, i₅) can be the normalized currents through bi-SQUID 20, (φ₁, φ₂,φ₃) can represent the phases across the Josephson junctions 12 a-12 c,β/2 is the parameter related to the inductance values and x_(e) (point14) can represent the point in the equations where the external fields(not shown) are included.

The equations for modeling the JJ's for the DC SQUID of FIG. 1 can bederived using Kirchhoff's current law, which is the principle ofconservation of electric charge and implies that at any junction in anelectrical circuit the sum of currents flowing into that node is equalto the sum of currents flowing out of that node, along with aresistively shunted junction (RSJ) model of the over-damped JJ. The JJ'scan be assumed to be symmetric such that for the normalized criticalcurrents i_(c1)=i_(c2)=1.0. The system of equations that models a singleDC SQUID dynamics is:

$\begin{matrix}{{{\overset{.}{\varphi}}_{1} = {J - {\frac{1}{\beta}\left( {\varphi_{1} - \varphi_{2} - \varphi_{e}} \right)} - {\sin\;\varphi_{1}}}}{{{\overset{.}{\varphi}}_{2} = {J + {\frac{1}{\beta}\left( {\varphi_{1} - \varphi_{2} - \varphi_{e}} \right)} - {\sin\;\varphi_{2}}}},}} & (1)\end{matrix}$where φ₁ and φ₂ are the phases across each of the Josephson junctionsand the dots denote the time differentiation with normalized time

τ = ω c ⁢ t = 2 ⁢ eI 0 ⁢ R N ⁢ t .The parameter ω_(c) in τ=2π/(Φ₀)(I₀)(R_(N)). The parameter R_(N) in τ isthe normal state resistance of the Josephson junctions and I₀ is thecritical current of the Josephson junctions.

$\beta = {2\pi\;\frac{{LI}_{0}}{\Phi_{0}}}$is the nonlinear parameter related to the SQUID inductance

$L,{J = \frac{I_{b}}{2I_{0}}},$where I_(b) is bias current, and φ_(e)=2πax_(e), where x_(e) is thenormalized external magnetic field and

$a = {{\beta.\mspace{14mu}\Phi_{0}} \equiv \frac{h}{2e} \approx {2.07 \times 10^{- 15}}}$tesla meter squared is the flux quantum, where h is Plank's constant and2e is the charge on the Cooper pair.

The phase equations for the single DC bi-SQUID schematic in FIG. 2 canbe derived in a similar way to those of the single DC SQUID. In thiscase there is a third junction J₃ that is related to the first andsecond junctions through the phases: φ₁+φ₃=φ₂. Using this relationship,all the terms that include φ₃ can be replaced with φ₂−φ₁, therebyreducing the number of phase equations needed to model the system fromthree to two. The governing equations for a single bi-SQUID are

$\begin{matrix}{{{\overset{.}{\varphi}}_{1} = {\frac{i_{b}}{2} - {\frac{1}{3\beta}\left( {\varphi_{1} - \varphi_{2} - \varphi_{e}} \right)} + {\frac{1}{3}i_{c\; 3}{\sin\left( {\varphi_{2} - \varphi_{1}} \right)}} - {\frac{2}{3}\sin\;\varphi_{1}} - {\frac{1}{3}\sin\;\varphi_{2}}}}{{{\overset{.}{\varphi}}_{2} = {\frac{i_{b}}{2} + {\frac{1}{3\beta}\left( {\varphi_{1} - \varphi_{2} - \varphi_{e}} \right)} - {\frac{1}{3}i_{c\; 3}{\sin\left( {\varphi_{2} - \varphi_{1}} \right)}} - {\frac{1}{3}\sin\;\varphi_{1}} - {\frac{2}{3}\sin\;\varphi_{2}}}},}} & (2)\end{matrix}$where

$i_{c\; 3} = \frac{I_{c\; 3}}{I_{0}}$is the normalized critical current on the third junction and all otherparameters are defined as the single DC SQUID.

The voltage response of both the single DC SQUID 10 of FIG. 1 and the DCbi-SQUID 20 of FIG. 2 can be simulated by integrating the systems ofequations that model the system dynamics, Eq (1) and Eq (2)respectively. After the phases φ₁ and φ₂ have been determined thederivatives {dot over (φ)}₁ and {dot over (φ)}₂ are evaluated and thetime-dependent voltage

${V(t)} = \frac{{\overset{.}{\varphi}}_{1} + {\overset{.}{\varphi}}_{2}}{2}$is calculated. The average voltage, <V>, of a SQUID (or bi-SQUID) at apoint in x_(e) is the mean value of the voltage over time. FIG. 3illustrates the results of this process. FIG. 3 is an average voltageresponse of a single DC SQUID compared with that of a single DCbi-SQUID. The average voltage response of the bi-SQUID (line 32), withthe proper selection of parameters, has a more linear average voltageresponse than the conventional DC SQUIDs (line 34 in FIG. 3) . Thesimulations were performed with J=1.0, β=1.0 and i_(c3)=1.0. The higherlinearity of the average voltage response of the bi-SQUID can increasethe overall utility of the device that incorporates the bi-SQUID as alinear amplifier, provided the bi-SQUIDs are incorporated to optimizethe overal linearity of the device.

The structure and cooperation of structure of the SQUID and bi-SQUID,and the design and performance of one-dimensional SQIF's using suchcomponents are discused in more detail in the aforementioned '994application, which has been incorporated in its entirety into thisspecification.

B. Dual Bi-SQUID Cell.

Referring now to FIG. 4, FIG. 4 depicts a circuit of a dual bi-SQUIDcell structure 40 of the present invention according to severalembodiments. As shown in FIG. 4, cell 40 can further include at leasttwo bi-SQUIDs 20 a, 20 b (indicated by the dotted lines in FIG. 4),which can be merged so that the bi-SQUIDs share at least one twobi-SQUID junctions 44 a, 44 b and at least one inductance 46. With thisconfiguration, cell 40 can present a diamond shaped appearance whenviewed in top plan. It should be appreciated, however, that othergeometric views (such as hexangonal, for example) are possible, providedthe equations discussed below are modified and derived using Kirchoff'slaw to account for the conservation of currents in such alternativeembodiments.

In FIG. 4, i_(b) can be taken to represent an input base current intocell 40, (i₁, . . . , i₁₁) can represent the normalized currents throughthe various bi-SQUID junctions 44, and (φ₁ . . . φ₆) can represent thephases across the JJ's 42 in cell 40 through bi-SQUIDs 20, (L₁, L_(2a),. . . , L_(6a), L_(2b), . . . , L_(6b)) represent the parameters relatedto the inductances and x_(e) (points 48 a, 48 b) can represent thepoints in the equations where the external fields are included. Usingthese representations, six governing equations are needed to simulatethe average voltage response for this device, which contains 12currents, 6 Josephson junctions and 9 inductors.

Kirchhoff's current law results in the following relations for thecurrents and phases in the diamond-shaped dual bi-SQUID

$\begin{matrix}\begin{matrix}{i_{b} = {i_{1} + i_{2}}} & {i_{1} = {i_{3} + i_{8}}} \\{{i_{2} + i_{3}} = i_{9}} & {i_{8} = {i_{7} + i_{10}}} \\{{i_{9} + i_{7}} = i_{11}} & {{i_{10} + i_{6}} = i_{4}} \\{i_{11} = {i_{6} + i_{5}}} & {{i_{4} + i_{5}} = i_{b}} \\{i_{1} = {{\sin\;\varphi_{1}} + {\overset{.}{\varphi}}_{1}}} & {i_{2} = {{\sin\;\varphi_{2}} + {\overset{.}{\varphi}}_{2}}} \\{i_{4} = {{\sin\;\varphi_{4}} + {\overset{.}{\varphi}}_{4}}} & {i_{5} = {{\sin\;\varphi_{5}} + {\overset{.}{\varphi}}_{5}}} \\{i_{3} = {{i_{c\; 3}\sin\;\varphi_{3}} + {\overset{.}{\varphi}}_{3}}} & {{i_{6} = {{i_{c\; 6}\sin\;\varphi_{6}} + {\overset{.}{\varphi}}_{6}}},}\end{matrix} & (3)\end{matrix}$where i_(c3)=I_(c3)/I₀ is the normalized critical current of the thirdjunction, i_(c6)=I_(c6)/I₀ is the normalized critical current of thesixth junction, I_(c1)=I_(c2)=I_(c4)=I_(c5)=I₀, and φ_(i) are the phasesacross of the Josephson junctions for i=1, . . . ,6. The dots on theequations in this specficiation denote time differentiation with respectto normalized time τ=ω_(c)t, where ω_(c) and t remain defined as above.The resistively-shunted junction (RSJ) model provides a current-phaserelation around the top half of the diamond of dual bi-SQUID cell 40(bi-SQUID 20 a)φ₁ +L ₁ i ₇ +L _(2a) i ₈=2πx _(e) a ₁+φ₂ +L _(2b) i ₉,where φ_(e)=2πa ₁ x _(e), where x_(e) is the normalized externalmagnetic flux and a₁=L₁+L_(2a)+L_(2b). Combining i₇=i₈−i₁₀ and i₇=i₁₁−i₉such that

${i_{7} = {\frac{i_{8} - i_{10}}{2} + \frac{i_{11} - i_{9}}{2}}},$and then substituting the current relations for i₇, i₁₀, i₉, and i₈yields

$\begin{matrix}{{{\left( {\frac{L_{1}}{2} + L_{2a}} \right)i_{1}} - {\frac{L_{1}}{2}i_{4}} + {\frac{L_{1}}{2}i_{5}} + {L_{1}i_{6}}} = {\varphi_{2} - \varphi_{1} + {\left( {\frac{L_{1}}{2} + L_{2b}} \right)i_{2}} + {2\pi\; x_{e}a_{1}} + {\left( {L_{1} + L_{2\; a} + L_{2b}} \right){i_{3}.}}}} & (4)\end{matrix}$

In order to get the first of the six equations that is needed todescribe the dynamics of the diamond shape, the bias current relationi₂=i_(b)−i₁ and Josephson junction relations from Eq. (3) aresubstituted into Eq. (4) to become

$\begin{matrix}{{{{L_{12}\left( {{\overset{.}{\varphi}}_{1} - {\overset{.}{\varphi}}_{3}} \right)} + {\frac{L_{1}}{2}\left( {{2{\overset{.}{\varphi}}_{6}} + {\overset{.}{\varphi}}_{5} - {\overset{.}{\varphi}}_{4}} \right)}} = {{\left( {\frac{L_{1}}{2} + L_{2b}} \right)i_{b}} + \varphi_{2} - \varphi_{1} + {2\pi\; x_{e}a_{1}} + {L_{12}\left( {{i_{c\; 3}\sin\;\varphi_{3}} - {\sin\;\varphi_{1}}} \right)} + {\frac{L_{1}}{2}\left( {{\sin\;\varphi_{4}} - {\sin\;\varphi_{5}} - {2\; i_{c\; 6}\sin\;\varphi_{6}}} \right)}}},} & (5)\end{matrix}$where L₁₂=L₁+L_(2a)+L_(2b).

To solve for the second of the six equations that governs the dynamicsof the single diamond the current relations for the bias currenti₁=i_(b)−i₂ and Josephson junctions from Eq. (3) are substituted intoEq. (4) to give

$\begin{matrix}{{{{\frac{L_{1}}{2}\left( {{2{\overset{.}{\varphi}}_{6}} + {\overset{.}{\varphi}}_{5} - {\overset{.}{\varphi}}_{4}} \right)} - {L_{12}\left( {{\overset{.}{\varphi}}_{2} + {\overset{.}{\varphi}}_{3}} \right)}} = {{{- \left( {\frac{L_{1}}{2} + L_{2\; a}} \right)}i_{b}} + \varphi_{2} - \varphi_{1} + {2\;\pi\; x_{e}a_{1}} + {L_{12}\left( {{i_{c\; 3}\sin\;\varphi_{3}} + {\sin\;\varphi_{2}}} \right)} + {\frac{L_{1}}{2}\left( {{\sin\;\varphi_{4}} - {\sin\;\varphi_{5}} - {2\; i_{c\; 6}\sin\;\varphi_{6}}} \right)}}},} & (6)\end{matrix}$where L₁₂=L₁+L_(2a)+L_(2b).

The RSJ model also provides a current-phase relation around the upperloop in the single diamond according to the equation φ₁+φ₃+L₃i₃=φ₂,where L₃=L_(3a)+L_(3b). Substituting the Josephson junction relationsfrom Eq. (3) and reorganizing yields the third of the six equations forthe dynamics of the single diamond.L ₃{dot over (φ)}₃=φ₂−φ₁−φ₃ −L ₃ i _(c3) sin φ₃.  (7)

A current-phase relation around the bottom half (bi-SQUID 20 b) of thesingle diamond cell 40 can be determined to beφ₄+L_(5a)i₁₀=2πx_(e)a₂+φ₅+L_(5b)i₁₁+L₁i₇, where φ_(e)=2πa₂x_(e) anda₂=L₁+L_(5a)+L_(5b). Substituting the current relations

${i_{7} = {\frac{i_{8} - i_{10}}{2} + \frac{i_{11} - i_{9}}{2}}},$i₁₀=i₄−i₆, i₁₁=i₅+i₆, i₉=i₂+i₃ and i₈=i₁−i₃ from Eq. (3) and thenreorganizing gives

$\begin{matrix}{{{\left( {L_{5\; a} + \frac{L_{1}}{2}} \right)i_{4}} - {\left( {L_{1} + L_{5\; a} + L_{5\; b}} \right)i_{6}}} = {\varphi_{5} - \varphi_{4} + {2\;\pi\; x_{e}a_{2}} + {\left( {L_{5\; b} + \frac{L_{1}}{2}} \right)i_{5}} + {\frac{L_{1}}{2}i_{1}} - {\frac{L_{1}}{2}i_{2}} - {L_{1}{i_{3}.}}}} & (8)\end{matrix}$Substituting the current relation for the bias current i₅=i_(b)−i₄ fromEq. (3) into Eq. (8) and then substituting the Josephson junctionrelations from Eq. (3) and reorganizing leads to the fourth governingequation

$\begin{matrix}{{{{L_{15}\left( {{\overset{.}{\varphi}}_{4} - {\overset{.}{\varphi}}_{6}} \right)} + {\frac{L_{1}}{2}\left( {{2{\overset{.}{\varphi}}_{3}} + {\overset{.}{\varphi}}_{2} - {\overset{.}{\varphi}}_{1}} \right)}} = {{\left( {L_{5\; b} + \frac{L_{1}}{2}} \right)i_{b}} + \varphi_{5} - \varphi_{4} + {2\;\pi\; x_{e}a_{2}} + {L_{15}\left( {{i_{c\; 6}\sin\;\varphi_{6}} - {\sin\;\varphi_{4}}} \right)} + {\frac{L_{1}}{2}\left( {{\sin\;\varphi_{1}} - {\sin\;\varphi_{2}} - {2\; i_{c\; 3}\sin\;\varphi_{3}}} \right)}}},} & (9)\end{matrix}$where L₁₅=L₁+L_(5a)+L_(5b).

To solve for the fifth of the six equations for the dynamics of thesingle diamond shape the current relations for the bias currenti₄=i_(b)−i₅ and Josephson junctions from Eq. (3) is substituted into Eq.(8) to become

$\begin{matrix}{{{{\frac{L_{1}}{2}\left( {{2{\overset{.}{\varphi}}_{3}} + {\overset{.}{\varphi}}_{2} - {\overset{.}{\varphi}}_{1}} \right)} - {L_{15}\left( {{\overset{.}{\varphi}}_{5} + {\overset{.}{\varphi}}_{6}} \right)}} = {{{- \left( {L_{5\; b} + \frac{L_{1}}{2}} \right)}i_{b}} + \varphi_{5} - \varphi_{4} + {2\pi\; x_{e}a_{2}} + {L_{15}\left( {{i_{c\; 6}\sin\;\varphi_{6}} + {\sin\;\varphi_{5}}} \right)} + {\frac{L_{1}}{2}\left( {{\sin\;\varphi_{1}} - {\sin\;\varphi_{2}} - {2i_{c\; 3}\sin\;\varphi_{3}}} \right)}}},} & (10)\end{matrix}$where L₁₅=L₁+L_(5a)+L_(5b).

For the sixth governing equations a current-phase relation can be foundfor the lower loop in the diamondφ₄+φ₆ +L ₆ i ₆=φ₅,where L₆=L_(6a)+L_(6b). Substituting the Josephson junction relationsfrom Eq. (3) into the relation and reorganizing yieldsL ₆{dot over (φ)}₆=φ₅−φ₄−φ₆ −L ₆ i _(c3) sin φ₃.  (11)

The equations from Eq. (5)-(7) and Eq. (9)-(11) can be combined toobtain the full system of equations that governs the phase dynamics ofthe diamond-shaped bi-SQUID:

$\begin{matrix}{{{{{L_{12}\left( {{\overset{.}{\varphi}}_{1} - {\overset{.}{\varphi}}_{3}} \right)} + {\frac{L_{1}}{2}\left( {{2{\overset{.}{\varphi}}_{6}} + {\overset{.}{\varphi}}_{5} - {\overset{.}{\varphi}}_{4}} \right)}} = {{\left( {\frac{L_{1}}{2} + L_{2\; b}} \right)i_{b}} + \varphi_{2} - \varphi_{1} + {2\pi\; x_{e}a_{1}} + {L_{12}\left( {{i_{c\; 3}\sin\;\varphi_{3}} - {\sin\;\varphi_{1}}} \right)} + {\frac{L_{1}}{2}\left( {{\sin\;\varphi_{4}} - {\sin\;\varphi_{5}} - {2\; i_{c\; 6}\sin\;\varphi_{6}}} \right)}}};}{{{{\frac{L_{1}}{2}\left( {{2{\overset{.}{\varphi}}_{6}} + {\overset{.}{\varphi}}_{5} - {\overset{.}{\varphi}}_{4}} \right)} - {L_{12}\left( {{\overset{.}{\varphi}}_{2} + {\overset{.}{\varphi}}_{3}} \right)}} = {{{- \left( {\frac{L_{1}}{2} + L_{2\; a}} \right)}i_{b}} + \varphi_{2} - \varphi_{1} + {2\pi\; x_{e}a_{1}} + {L_{12}\left( {{i_{c\; 3}\sin\;\varphi_{3}} + {\sin\;\varphi_{2}}} \right)} + {\frac{L_{1}}{2}\left( {{\sin\;\varphi_{4}} - {\sin\;\varphi_{5}} - {2\; i_{c\; 6}\sin\;\varphi_{6}}} \right)}}};}{{{L_{3}{\overset{.}{\varphi}}_{3}} = {\varphi_{2} - \varphi_{1} - \varphi_{3} - {L_{3}i_{c\; 3}\sin\;\varphi_{3}}}};}{{{{L_{15}\left( {{\overset{.}{\varphi}}_{4} - {\overset{.}{\varphi}}_{6}} \right)} + {\frac{L_{1}}{2}\left( {{2{\overset{.}{\varphi}}_{3}} + {\overset{.}{\varphi}}_{2} - {\overset{.}{\varphi}}_{1}} \right)}} = {{\left( {L_{5\; b} + \frac{L_{1}}{2}} \right)i_{b}} + \varphi_{5} - \varphi_{4} + {2\pi\; x_{e}a_{2}} + {L_{15}\left( {{i_{c\; 6}\sin\;\varphi_{6}} - {\sin\;\varphi_{4}}} \right)} + {\frac{L_{1}}{2}\left( {{\sin\;\varphi_{1}} - {\sin\;\varphi_{2}} - {2\; i_{c\; 3}\sin\;\varphi_{3}}} \right)}}};}{{{{{\frac{L_{1}}{2}\left( {{2{\overset{.}{\varphi}}_{3}} + {\overset{.}{\varphi}}_{2} - {\overset{.}{\varphi}}_{1}} \right)} - {L_{15}\left( {{\overset{.}{\varphi}}_{5} + {\overset{.}{\varphi}}_{6}} \right)}} = {{{- \left( {L_{5b} + \frac{L_{1}}{2}} \right)}i_{b}} + \varphi_{5} - \varphi_{4} + {2\pi\; x_{e}a_{2}} + {L_{15}\left( {{i_{c\; 6}\sin\;\varphi_{6}} + {\sin\;\varphi_{5}}} \right)} + {\frac{L_{1}}{2}\left( {{\sin\;\varphi_{1}} - {\sin\;\varphi_{2}} - {2\; i_{c\; 3}\sin\;\varphi_{3}}} \right)}}};{and}},{{L_{6}{\overset{.}{\varphi}}_{6}} = {\varphi_{5} - \varphi_{4} - \varphi_{6} - {L_{6}i_{c\; 3}\sin\;\varphi_{3}}}},}} & (12)\end{matrix}$where L₁₂=L₁+L_(2a)+L_(2b), L₃=L_(3a)+L_(3b), L₁₅=L₁+L_(5a)+L_(5b),L₆=L_(6a)+L_(6b) and φ_(i) are the phases on each of the Josephsonjunctions, i=1, . . . ,6. The normalized critical current on the thirdjunction is

${i_{c\; 3} = \frac{I_{c\; 3}}{I_{0}}},$the normalized critical current of the sixth junction is

${i_{c\; 6} = \frac{I_{c\; 6}}{I_{0}}},$I_(c1)=I_(c2)=I_(c4)=I_(c5)=I₀, x_(e) is the normalized externalmagnetic flux, a₁=L₁+L_(2a)+L_(2b) and a₂=L₁+L_(5a)+L_(5b).

Once the equations above are solved for inductances, values can besubstituted there for the inductance and the average voltage responsefor the diamond shaped bi-SQUID cell 40 of the present inventionaccording to several embodiments is shown in FIG. 5. The response can beplottted for J=1.001, L₁=0.54, L_(2a)=L_(2b)=L_(5a)=L_(5b)=0.24,L_(3a)=L_(3b)=L_(6a)=L_(6b)=0.3, and i_(c3)=0.5 These values are allnormalized values, without units. As shown in FIG. 5, this averagevoltage response retains its linearity, with proper choice ofparameters, such as varying the inductances instead of the loop sizesthe triangle-shaped bi-SQUIDs 20 in the cell. The actual size of thetriangle could also be varied.

Referring now to FIG. 6, FIG. 6 is a microphotograph of the singlediamond bi-SQUID of FIG. 4. FIG. 6 shows the implementation of thediamond-shaped dual bi-SQUID cell once fabricated. One such fabricationmethod that can be used is the HYPRES Nb fabrication process, asdescribed in Niobium (Nb) process design rule, revision #24, Jan. 11,2008, 11, available at http://www.hypres.com. R_(shunt) is the shuntingresistance and V_(c) is the critical voltage across each of the JJ's.For the fabrication shown in FIG. 6, J1=J2=J3=0.25 mA, R_(shunt)=2.4 Ω,V_(c)=I_(c)R_(sh)=600 μV.

The cell 40 shown in FIG. 6 can be manufactured using all four Nblayers: a ground plane layer, two layers for junctions and inductors,and a top layer to implement a flux bias line 50 overlaying the bi-SQUIDcells. The ground plane was used only under Nb layers forming bi-SQUIDinductors and junctions in order to maintain their low specificinductance. However, ground plane can be partially removed from underinductor 46 to increase its inductance value, if desired. The groundplane can also be removed from the central area of the bi-SQUID loops(areas 52 in FIG. 6) to allow an external magnetic field to threadthrough the cell.

C. Implementation of 2D Diamond bi-SQUID SQIF Arrays.

1. Construction of 2D SQIF Array.

To construct the 2D SQIF array of the present invention according toseveral embodiments, and referring now to FIGS. 7-9, building blocks ofdual bi-SQUID cells 40 like those shown in FIG. 6 can be used. Therepeating pattern in the 2D array that is illustrated in FIGS. 7-9 canbe a series of cells 40 having a diamond shape created by two bi-SQUIDs20 a, 20 b, which can have the structure as recited above for cell 40(See FIG. 4). An example of such as array is shown electrically and intop plan in FIG. 7. The phase equations that model the 2D diamond arrayscan be derived in a similar way as the single diamond structure ofbi-SQUIDs. But the above analysis in the previous section and Equations(5)-(7) and (9)-(11) only models the response for one cell 40. To modelthe effects for the SQIF array as a whole, the analysis can now beextended to the full 2D diamond array. But because of the tightly packedarrangement of diamond-shaped cells 40, each bi-SQUID shares its JJs andinductances with neighboring bi-SQUIDs. Stated differently, eachbi-SQUID cell 40 affect adjacent bi-SQUID cells 40 as well of thebi-SQUIDs of other cells 40 in the array 70. Therefore to model theentire array, Kirchoff's law can be used to sum current thorugh each andevery junciton for a network of cells 40 that share inductances, insteadof a single cell. Also, the effects that each cell 40 has on every othercell in the 2D array, and the effects that occur at boundary conditionscan be accounted for (again, using Kirchoff's law) in the methods of thepresent invention.

At the corners of the 2D SQIF array 70, each dual bi-SQUID cell 40 has acontact to three neighboring cells at cell junctions 80. FIGS. 8b, 8d,8g and 8i illustrate this boundary condition at the respective upperleft, upper right, lower left and lower right corners of the 2D SQIFarray. FIGS. 8e and 8f illustrate the boundary conditions on therespective left side and right side of the 2D SQIF array. FIGS. 8c and8h illustrate the respective top and bottom conditions on the 2D SQIFarray. The array dc bias can be fed uniformly from the top bi-SQUID row(cells 40 _(1,1) through 40 _(1,N) of the 2D SQIF array 70). Similarly,the array can be grounded to the bottom bi-SQUID row (cells 40 _(M,1)through 40 _(M,N) or array 70 in FIG. 7). The inductively coupled fluxbias line is overlaid on the top of the array forming loops for eachcolumn. The direction of the de flux bias control current is from top tobottom of array 70 in FIG. 7. The 2-D array design in FIGS. 7-8 avoidsthe use of long parasitic wires, as every component of the array is anessential element of a bi-SQUID, which results in an efficient use ofthe available area.

To account for the boundary conditions in an M×N 2D array, and referringagain to FIG. 7, The number of diamonds cells 40 in the verticaldirection can be 2M and the number of bi-SQUIDs in the horizontaldirection can be 2N-1. This means that, for example a 10×8 array canhave 20×15=300 bi-SQUIDs. The segmenting of the array in the mannerallows for the smallest repeated pattern grouping, which can furtherresult in the simplest set of modeling equations for computationalpurposes. The 2D array equations are derived in pieces that include thediamond structure and two transition areas, as shown by area 82 in FIG.8a . This segmentation of the array is the smallest repeatable pattern.

Using the above methodology, the equation for the 2D array cells in theinterior of the array, i.e. cells 40 _(j,i) where j=2 to M-1 and i=2 toN-1 as shown in FIG. 8a results in the following equations (13) through(20) in Appendix found at the end of this Specification. Next, themethodology was used for the top row of cells 40 _(j,i) for j=1 and i=2,. . . , N-1 (FIG. 8c ) resulting in the development of equations (21)through (28) of the Appendix. Still further, the methodology was used todevelop the equations (29) through (36), which model the bottom row ofcells 40 _(j,i) for j=M and i=2, . . . ,N-1 (FIG. 8h ). In similarfashion, the equations that model the left side cells 40 _(j,i) for j=2,. . . , M-1 and i=1 (Equations (37) through (44), FIG. 8e ), the topleft hand corner cell 40 _(1,1) (FIG. 8b and Equations (45) through(52)), the bottom left hand corner cell 40 _(M,1) (FIG. 8g , AppendixEquations (53) through (60)) were developed. Finally, the equations forthe rights side of array 70, cells 40 _(j,i) for j=2, . . . , M-1 andi=N of the array (FIG. 8f ) were developed and listed as Equations (61)through (66) in the equations Appendix, and the equations for the topright hand corner cell 40 _(1,N) (FIG. 8d , Equations (67) through(72)), and bottom right hand corner cell 40 _(M,N) (FIG. 8i , Equations(73) through (78)) were developed. The net result is a matrix ofsixty-six equations, Equations (13) through (78), which can accuratelythe behavior of the 2D SQIF array.

As described above for a single cell 40, once Equations (13)-(78) aredeveloped and solved for phases φ and phase rates (derivatives of φ) foreach JJ in each cell 40. These derivatives are used to calculate theaverage voltage response of the bi-SQUID array, which is the outputmeasured from the fabricated bi-SQUID chips Next, the cell can bemanufactured with the cell inductances that achieve the desired phaseand phase rate (and by extension, the desired voltage response). Theinductances, critical currents, normal state resistance, bias currentsand external field are known or controllable. One way this has beenaccomplished in the prior art was by varying the loop size of bi-SQUID20. But for SQIFarray 70, uniform loop sizes are needed for ease ofmanufacture of diamon cells 40 and 2D SQIF array 70. So, in order tovary the inductances in 2D SQIF array 70 while the a uniform loop size(area of the cell 40), the inductances, critical currents, normal stateresistance, bias currents and external field are known or controllable.The arrays can be constructed by varying the area of triangle arraycells 20 and ground plane hole sizes in a manner that reults in a normalGaussian distribution of cell inductances.

FIG. 9 shows layout implementation of a portion of the 80×15 2D array ofFIG. 7, using diamond dual bi-SQUID cells. The total area of the 2Darray of dual-bi-SQUID cells shown in for an 80×15 2D array can be 1623μm². The central area of each cell 40 contains ground plane openings tofacilitate RF signal reception.

2. Experimental Investigation 2D SQIF Noise Properties

FIG. 10 illustrates the measured flux-to-voltage characteristics of a15×80 2D SQIF array of cell 40 of FIGS. 7-9, with a 70% Gaussian spread(σ=70%) of inductance values. Like FIG. 3 for SQUIDs and bi-SQUIDs, FIG.10 is a graph of the average voltage, <V>, of the 2D SQIF array at apoint in x_(e), which is the mean value of the voltage over time. Asshown in FIG. 10, the measured flux/voltage characteristic for σ=70% is50 mV/div 0.5 mA/div, max voltage=295 mV, ΔV/ΔI (flux bias) ˜735 V/A.FIGS. 11-13 show the corresponding flux noise measurement at themid-point of the positive slope of the anti-peak (point 1002 of peak1000 in FIG. 10). As can be seen from FIG. 11, the flux noise spectraldensity can approach 2×10⁻⁶Φ₀/√{square root over (Hz)}, which is theexpected value for this size array. FIGS. 12 and 13 show thecorresponding energy characteristics of the (15×80) bi-SQUID SQIF array,which were calculated as follows:

Noise energy per unit bandwidth via flux noise in a SQUID:

$\begin{matrix}{{{ɛ(f)} = \frac{S_{\Phi}(f)}{2\; L}},} & (79)\end{matrix}$where f is frequency, L−inductance of bi-SQUID calculated from themeasured seperately ΔI_(c) modulations of the curve defined as

${L = \frac{\Phi_{0}}{\Delta\; I_{c}}};$and,

Noise temperature:

$\begin{matrix}{T_{N} = \frac{\pi\; f\;{ɛ(f)}}{k_{B}}} & (80)\end{matrix}$where k_(B) is Boltzmann's constant.

As shown by FIGS. 10-13, a well-defined zero-field anti-peak for a 2DSQIF array of cells 40 can be maintained if the mutual inductancesbetween cells and the boundary conditions are accounted for. Stateddifferently, more bi-SQUID's can be packed into a corresponding areasusing the methods of the present invention without sacrificing theoverall linearity anti-peak response of the 2D SQIF array, whenconsidered as a whole.

3. 2D SQIF Array Antenna Sensitivity Analysis

The sensitivities illustrated in FIGS. 10-13 were based on a SQIF-basedantenna, by assuming that the area for a single diamond shaped (double)bi-SQUID cell 40 is 1.62×10⁻⁹ m². Using that as the effective area forcell 40, multiplying by 15×80/2 to account for the number of individualbi-SQUIDs in the array, and taking the flux noise from FIG. 8(b) as2×10⁻⁶Φ₀/√{square root over (Hz)}, gives a field sensitivity of 4.25fT/√{square root over (Hz)}. Assuming a scaling as a function of√{square root over (N)}, the field noise for a 1000×2000 array would be0.104 fT/√{square root over (Hz)} at 100 kHz. Approximating the physicaldimension of the diamond cell 40 as 71 μm×71 μm means that the diamondoccupies an area of 5×10⁻⁹ m² (consistent with the effective area of thediamond shaped bi-SQUID). Thus a 1000×2000 array would occupy an area of50 cm², since each diamond contains two bi-SQUIDs. This corresponds to asquare with 7.1 cm on a side. A corresponding linear (1D) array of theprior art, which would essentially be 2,000,000 bi-SQUIDs that aremeandered in series, would require a much larger area, an area of 10,562cm².

Referring now to FIG. 14, a block diagram 100 that illustrates stepsthat can be taken to practice the methods of the present inventionaccording to several embodiments is shown. As shown method 100 caninclude the initial step 102 of providing a plurality of bi-SQUID's 20.The bi-SQUIDs 20 can have the structure as described above. The methodscan further include the step 104 of merging pairs of bi-SQUID's 20 intodiamond-shaped cells 40 and connecting the ells so that each cell 40shares at least three cell junctions 80 (step 106). Next, the methodsaccording to several embodiments can include the step 108 of modelingthe phases of the 2D SQIF array. Step 108 can further be accomplished byaccounting for mutual inductances between cells 40 (step 110 in FIG. 14)and by using the equations (13) through (78) of the Appendix to modelthe boundarys conditions of array 70 (step 112 in FIG. 14). As shown bystep 114 in FIG. 14, the methods according to several embodiments canfurther include the step of manufacturing cells 40 with inductances thatachieve the phases and phase behavior resulting form the accomplishmentof step 108, in order the achieve the anti-peak response shown in FIG.10. For embodiments the loop size of cell 40 is uniform, one way to dothis can be to vary the ground plane hole size, as discussed above.

The use of the terms “a” and “an” and “the” and similar references inthe context of describing the invention (especially in the context ofthe following claims) is to be construed to cover both the singular andthe plural, unless otherwise indicated herein or clearly contradicted bycontext. The terms “comprising,” “having,” “including,” and “containing”are to be construed as open-ended terms (i.e., meaning “including, butnot limited to,”) unless otherwise noted. Recitation of ranges of valuesherein are merely intended to serve as a shorthand method of referringindividually to each separate value falling within the range, unlessotherwise indicated herein, and each separate value is incorporated intothe specification as if it were individually recited herein. All methodsdescribed herein can be performed in any suitable order unless otherwiseindicated herein or otherwise clearly contradicted by context. The useof any and all examples, or exemplary language (e.g., “such as”)provided herein, is intended merely to better illuminate the inventionand does not pose a limitation on the scope of the invention unlessotherwise claimed. No language in the specification should be construedas indicating any non-claimed element as essential to the practice ofthe invention.

Preferred embodiments of this invention are described herein, includingthe best mode known to the inventors for carrying out the invention.Variations of those preferred embodiments may become apparent to thoseof ordinary skill in the art upon reading the foregoing description. Theinventors expect skilled artisans to employ such variations asappropriate, and the inventors intend for the invention to be practicedotherwise than as specifically described herein. Accordingly, thisinvention includes all modifications and equivalents of the subjectmatter recited in the claims appended hereto as permitted by applicablelaw. Moreover, any combination of the above-described elements in allpossible variations thereof is encompassed by the invention unlessotherwise indicated herein or otherwise clearly contradicted by context.

APPENDIX 2D Diamond Array Governing Equations

For the model equations below, the variables φ_(m,i,j), m=1, . . . ,8,i=1, . . . , N, j=1, . . . , M are the phases on each of the Josephsonjunctions (m=1, . . . ,8). The dots denote the time differentiation withnormalized time τ=Ω_(c)t, where t is time and

ω c = 2 ⁢ eI c ⁢ R Nis the normalized time constant. The parameter R_(N) in Ω_(c) is thenormal state resistance of the Josephson junctions, I_(c) is thecritical current of the Josephson junctions, e is the charge of anelectron, and h is the reduced Planck constant.

$i_{b} = \frac{I_{b}}{I_{c}}$is the normalized bias current, where I_(b) is the bias current.

$\Phi_{0} \equiv \frac{h}{2e} \approx {2.07 \times 10^{- 15}}$tesla meter squared is the flux quantum, where h is Plank's constant and2e is the charge on the Cooper pair. The normalized critical current onthe third junction is

${i_{{c\; 3},i,j} = \frac{I_{{c\; 3},i,j}}{I_{c}}},$the normalized critical current of the sixth junction is

${i_{{c\; 6},i,j} = \frac{I_{{c\; 6},i,j}}{I_{c}}},$the normalized critical current of the seventh junction is

${i_{{c\; 7},i,j} = \frac{I_{{c\; 7},i,j}}{I_{c}}},$the normalized critical current of the eighth junction is

${i_{{c\; 8},i,j} = \frac{I_{{c\; 8},i,j}}{I_{c}}},$I_(c1,i,j)=I_(c2,i,j)=I_(c4,i,j)=I_(c5,i,j)=I₀,

$x_{e,i,j} = \frac{B_{e}}{\Phi_{0}}$is the normalized external magnetic flux per unit area and a_(n,i,j),n=1, . . . ,4 is the bi-SQUID area. The approximate assumptions thata_(1,i,j)=L_(1,i,j)+L_(2a,i,j)+L_(2b,i,j),a₂=L_(1,i,j)+L_(5a,i,j)+L_(5b,i,j),a_(3,i,j)=L_(4,i,j)+L_(2a,i+1,j)+L_(2b,i,j),a₄=L_(4,i,j+1)+L_(5a,i+1,j)+L_(5b,i,j) are used.

Using the assumptions and variables able, the equations for the array 70interior cells, i.e, Cells 40 _(j,i) for j=2, . . . ,M-1 and i=2, . . .,N-1 become (8 equations):

$\begin{matrix}{{{L_{{2a},i,j}\left( {{\overset{.}{\varphi}}_{1,i,j} + {\overset{.}{\varphi}}_{1,{i - 1},j} - {\overset{.}{\varphi}}_{3,i,j}} \right)} - {L_{{2b},i,j}\left( {{\overset{.}{\varphi}}_{2,i,j} + {\overset{.}{\varphi}}_{3,i,j} - {\overset{.}{\varphi}}_{7,i,j}} \right)} + {L_{1,i,j}\left( {{\sum\limits_{k = 1}^{i}{\overset{.}{\varphi}}_{1,k,j}} + {\sum\limits_{k = 1}^{i - 1}{\overset{.}{\varphi}}_{2,k,j}} - {\overset{.}{\varphi}}_{3,i,j} - {\sum\limits_{k = 1}^{i}{\overset{.}{\varphi}}_{4,k,j}} - {\sum\limits_{k = 1}^{i - 1}{\overset{.}{\varphi}}_{5,k,j}} - {\overset{.}{\varphi}}_{6,i,j}} \right)}} = {\varphi_{2,i,j} - \varphi_{1,i,j} + {2\pi\; x_{e}a_{1,i,j}} - {L_{{2a},i,j}\left( {{\sin\;\varphi_{1,i,j}} + {i_{{c\; 7},{i - 1},j}\sin\;\varphi_{7,{i - 1},j}} - {i_{{c\; 3},i,j}\sin\;\varphi_{3,i,j}}} \right)} + {L_{{2b},i,j}\left( {{\sin\;\varphi_{2,i,j}} + {i_{{c\; 3},i,j}\sin\;\varphi_{3,i,j}} - {i_{{c\; 7},i,j}\sin\;\varphi_{7,i,j}}} \right)} - {L_{1,i,j}\left( {{\sum\limits_{k = 1}^{i}{\sin\;\varphi_{1,k,j}}} + {\sum\limits_{k = 1}^{i - 1}{\sin\;\varphi_{2,k,j}}} - {i_{{c\; 3},i,j}\sin\;\varphi_{3,i,j}} - {\sum\limits_{k = 1}^{i}{\sin\;\varphi_{4,k,j}}} - {\sum\limits_{k = 1}^{i - 1}{\sin\;\varphi_{5,k,j}}} - {i_{{c\; 6},i,j}\sin\;\varphi_{6,i,j}}} \right)}}} & {{Equation}\mspace{14mu}(13)}\end{matrix}$

$\begin{matrix}{{{L_{{2b},i,j}\left( {{\overset{.}{\varphi}}_{2,i,j} + {\overset{.}{\varphi}}_{3,i,j} - {\overset{.}{\varphi}}_{7,i,j}} \right)} - {L_{{2a},i,j}\left( {{\overset{.}{\varphi}}_{1,{i + 1},j} + {\overset{.}{\varphi}}_{7,i,j} - {\overset{.}{\varphi}}_{3,{i + 1},j}} \right)} - {L_{4,i,j}\left( {{\sum\limits_{k = 1}^{i}{\overset{.}{\varphi}}_{4,k,{j - 1}}} + {\sum\limits_{k = 1}^{i}{\overset{.}{\varphi}}_{5,k,{j - 1}}} - {\sum\limits_{k = 1}^{i}{\overset{.}{\varphi}}_{1,k,j}} - {\sum\limits_{k = 1}^{i}{\overset{.}{\varphi}}_{2,k,j}}} \right)}} = {\varphi_{1,{i + 1},j} - \varphi_{2,i,j} + {2\pi\; x_{e}a_{3,i,j}} - {L_{{2b},i,j}\left( {{\sin\;\varphi_{2,i,j}} + {i_{{c\; 3},i,j}\sin\;\varphi_{3,i,j}} - {i_{{c\; 7},i,j}\sin\;\varphi_{7,i,j}}} \right)} + {L_{{2\; a},{i + 1},j}\left( {{\sin\;\varphi_{1,{i + 1},j}} + {i_{{c\; 7},i,j}\sin_{7,i,j}} - {i_{{c\; 3},{i + 1},j}\sin\;\varphi_{3,{i + 1},j}}} \right)} + {L_{4,i,j}\left( {{\sum\limits_{k = 1}^{i}{\sin\;\varphi_{4,k,{j - 1}}}} + {\sum\limits_{k = 1}^{i}{\sin\;\varphi_{5,k,{j - 1}}}} - {\sum\limits_{k = 1}^{i}{\sin\;\varphi_{1,k,j}}} - {\sum\limits_{k = 1}^{i}{\sin\;\varphi_{2,k,j}}}} \right)}}} & {{Equation}\mspace{14mu}(14)}\end{matrix}$(L _(3a,i,j) +L _(3b,i,j)){dot over(φ)}_(3,i,j)=φ_(2,i,j)−φ_(1,i,j)−φ_(3,i,j)−(L _(3a,i,j) +L _(3b,i,j))i_(c3,i,j) sin φ_(3,i,j)  Equation (15)

$\begin{matrix}{{{L_{{5a},i,j}\left( {{\overset{.}{\varphi}}_{4,i,j} + {\overset{.}{\varphi}}_{6,i,j} - {\overset{.}{\varphi}}_{8,{i - 1},j}} \right)} - {L_{{5b},i,j}\left( {{\overset{.}{\varphi}}_{8,i,j} + {\overset{.}{\varphi}}_{5,i,j} - {\overset{.}{\varphi}}_{6,i,j}} \right)} - {L_{1,i,j}\left( {{\sum\limits_{k = 1}^{i}{\overset{.}{\varphi}}_{1,k,j}} + {\sum\limits_{k = 1}^{i - 1}{\overset{.}{\varphi}}_{2,k,j}} - {\overset{.}{\varphi}}_{3,i,j} - {\sum\limits_{k = 1}^{i}{\overset{.}{\varphi}}_{4,k,j}} - {\sum\limits_{k = 1}^{i - 1}{\overset{.}{\varphi}}_{5,k,j}} - {\overset{.}{\varphi}}_{6,i,j}} \right)}} = {\varphi_{5,i,j} - \varphi_{4,i,j} + {2\pi\; x_{e}a_{2,i,j}} - {L_{{2a},i,j}\left( {{\sin\;\varphi_{4,i,j}} + {i_{{c\; 6},i,j}\sin\;\varphi_{6,i,j}} - {i_{{c\; 8},{i - 1},j}\sin\;\varphi_{8,{i - 1},j}}} \right)} + {L_{{5b},i,j}\left( {{\sin\;\varphi_{5,i,j}} + {i_{{c\; 8},i,j}\sin\;\varphi_{{8.i},j}} - {i_{{c\; 6},i,j}\sin\;\varphi_{6,i,j}}} \right)} + {L_{1,i,j}\left( {{\sum\limits_{k = 1}^{i}{\sin\;\varphi_{1,k,j}}} + {\sum\limits_{k = 1}^{i - 1}{\sin\;\varphi_{2,k,j}}} - {i_{{c\; 3},i,j}\sin\;\varphi_{3,\; i,j}} - {\sum\limits_{k = 1}^{i}{\sin\;\varphi_{4,k,j}}} - {\sum\limits_{k = 1}^{i - 1}{\sin\;\varphi_{5,{k.j}}}} - {i_{{c\; 6},i,j}\sin\;\varphi_{6,i,j}}} \right)}}} & {{Equation}\mspace{14mu}(16)}\end{matrix}$

$\begin{matrix}{{{L_{{5b},i,j}\left( {{\overset{.}{\varphi}}_{5,i,j} + {\overset{.}{\varphi}}_{8,i,j} - {\overset{.}{\varphi}}_{6,i,j}} \right)} - {L_{{5a},{i + 1},j}\left( {{\overset{.}{\varphi}}_{4,{i + 1},j} + {\overset{.}{\varphi}}_{6,{i + 1},j} - {\overset{.}{\varphi}}_{8,i,j}} \right)} + {L_{4,i,{j + 1}}\left( {{\sum\limits_{k = 1}^{i}{\overset{.}{\varphi}}_{4,k,j}} + {\sum\limits_{k = 1}^{i}{\overset{.}{\varphi}}_{5,k,j}} - {\sum\limits_{k = 1}^{i}{\overset{.}{\varphi}}_{1,k,{j + 1}}} - {\sum\limits_{k = 1}^{i}{\overset{.}{\varphi}}_{2,k,{j + 1}}}} \right)}} = {\varphi_{4,{i + 1},j} - \varphi_{5,i,j} + {2\pi\; x_{e}a_{4,i,j}} - {L_{{5b},i,j}\left( {{\sin\;\varphi_{5,i,j}} + {i_{{c\; 8},i,j}\sin\;\varphi_{8,i,j}} - {i_{{c\; 6},i,j}\sin\;\varphi_{6,i,j}}} \right)} + {L_{{5a},{i + 1},j}\left( {{\sin\;\varphi_{4,{i + 1},j}} + {i_{{c\; 6},{i + 1},j}\sin\;\varphi_{6,{i + 1},j}} - {i_{{c\; 8},i,j}\sin\;\varphi_{8,i,j}}} \right)} - {L_{4,i,{j + 1}}\left( {{\sum\limits_{k = 1}^{i}{\sin\;\varphi_{4,k,j}}} + {\sum\limits_{k = 1}^{i}{\sin\;\varphi_{5,k,j}}} - {\sum\limits_{k = 1}^{i}{\sin\;\varphi_{1,k,{j + 1}}}} - {\sum\limits_{k = 1}^{i}{\sin\;\varphi_{2,k,{j + 1}}}}} \right)}}} & {{Equation}\mspace{14mu}(17)}\end{matrix}$(L _(6a,i,j) +L _(6b,i,j)){dot over(φ)}_(6,i,j)=φ_(4,i,j)−φ_(5,i,j)−φ_(6,i,j)−(L _(6a,i,j) +L _(6b,i,j))i_(c3,i,j) sin φ_(3,i,j)  Equation (18)(L _(7a,i,j) +L _(7b,i,j)){dot over (φ)}_(7,i,j) +L _(2a,i+1,j)({dotover (φ)}_(1,i+1,j)+{dot over (φ)}_(7,i,j)−{dot over (φ)}_(3,i+1,j))−L_(2b,i,j)({dot over (φ)}_(2,i,j)+{dot over (φ)}_(3,i,j)−{dot over(φ)}_(7,i,j))=−φ_(7,i,j)−(L _(7a,i,j) +L _(7b,i,j))i _(c7,i,j) sinφ_(7,i,j) −L _(2a,i+1,j)(sin φ_(1,i+1,j) +i _(c7,i,j) sin φ_(7,i,j) −i_(c3,i+1,j) sin φ_(3,i+1,j))+L _(2b,i,j)(sin φ_(2,i,j) +i _(c3,i,j) sinφ_(3,i,j) −i _(c7,i,j) sin φ_(7,i,j))  Equation (19)(L _(8a,i,j) +L _(8b,i,j)){dot over (φ)}_(8,i,j) +L _(5b,i,j)({dot over(φ)}_(5,i,j)+{dot over (φ)}_(8,i,j)−{dot over (φ)}_(6,i,j))−L_(5b,i+1,j)({dot over (φ)}_(4,i+1,j)+{dot over (φ)}_(6,i+1,j)−{dot over(φ)}_(8,i,j))=−φ_(8,i,j)−(L _(8a,i,j) +L _(8b,i,j))i _(c8,i,j) sinφ_(7,i,j) −L _(5b,i,j)(sin φ_(5,i,j) +i _(c8,i,j) sin φ_(8,i,j) −i_(c6,i,j) sin φ_(6,i,j))+L _(5a,i+1,j)(sin φ_(4,i+1,j) +i _(c6,i+1,j)sin φ_(6,i+1,j) −i _(c8,i,j) sin φ_(8,i,j))  Equation (20)

For the cells in the top row of array 70, i.e. for cells 40 _(j,i) forj=1 and i=2, . . . , N-1 (8 equations):

$\begin{matrix}{{{L_{{2a},i,j}\left( {{\overset{.}{\varphi}}_{1,i,1} + {\overset{.}{\varphi}}_{7,{i - 1},1} - {\overset{.}{\varphi}}_{3,i,1}} \right)} - {L_{{2b},i,1}\left( {{\overset{.}{\varphi}}_{2,i,1} + {\overset{.}{\varphi}}_{3,i,1} - {\overset{.}{\varphi}}_{7,i,1}} \right)} + {L_{1,i,1}\left( {{\sum\limits_{k = 1}^{i}{\overset{.}{\varphi}}_{1,k,1}} + {\sum\limits_{k = 1}^{i - 1}{\overset{.}{\varphi}}_{2,k,1}} - {\overset{.}{\varphi}}_{3,i,1} - {\sum\limits_{k = 1}^{i}{\overset{.}{\varphi}}_{4,k,1}} - {\sum\limits_{k = 1}^{i - 1}{\overset{.}{\varphi}}_{5,k,1}} - {\overset{.}{\varphi}}_{6,i,1}} \right)}} = {\varphi_{2,i,1} - \varphi_{1,i,1} + {2\pi\; x_{e}a_{1,i,1}} - {L_{{2a},i,1}\left( {{\sin\;\varphi_{1,i,1}} + {i_{{c\; 7},{i - 1},i}\sin\;\varphi_{7,{i - 1},1}} - {i_{{c\; 3},i,1}\sin\;\varphi_{3,i,1}}} \right)} + {L_{{2b},i,1}\left( {{\sin\;\varphi_{2,i,1}} + {i_{{c\; 3},i,1}\sin\;\varphi_{3,i,1}} - {i_{{c\; 7},i,1}\sin\;\varphi_{7,i,1}}} \right)} - {L_{1,i,1}\left( {{\sum\limits_{k = 1}^{i}{\sin\;\varphi_{1,k\;,1}}} + {\sum\limits_{k = 1}^{i - 1}{\sin\;\varphi_{2,k,1}}} - {i_{{c\; 3},i,1}\sin\;\varphi_{3,i,1}} - {\sum\limits_{k = 1}^{i}{\sin\;\varphi_{4,k,1}}} - {\sum\limits_{k = 1}^{i - 1}{\sin\;\varphi_{5,k,1}}} - {i_{{c\; 6},i,1}\sin\;\varphi_{6,i\;,1}}} \right)}}} & {{Equation}\mspace{14mu}(21)}\end{matrix}$

$\begin{matrix}{{{L_{{2b},i,1}\left( {{\overset{.}{\varphi}}_{2,i,1} + {\overset{.}{\varphi}}_{3,i,1} - {\overset{.}{\varphi}}_{7,i,1}} \right)} - {L_{{2a},{i + 1},1}\left( {{\overset{.}{\varphi}}_{1,{i + 1},1} + {\overset{.}{\varphi}}_{7,i,1} - {\overset{.}{\varphi}}_{3,{i + 1},1}} \right)} + {L_{4,i,1}\left( {{\sum\limits_{k = 1}^{i}{\overset{.}{\varphi}}_{1,k,1}} + {\sum\limits_{k = 1}^{i}{\overset{.}{\varphi}}_{2,k,1}}} \right)}} = {{i \times L_{4,i,1}i_{b}} + \varphi_{1,{i + 1},1} - \varphi_{2,i,1} + {2\pi\; x_{e}a_{{3f},i,1}} - {L_{{2b},i,1}\left( {{\sin\;\varphi_{2,i,1}} + {i_{{c\; 3},i,1}\sin\;\varphi_{3,i,1}} - {i_{{c\; 7},i,1}\sin\;\varphi_{7,i,1}}} \right)} + {L_{{2a},{i + 1},1}\left( {{\sin\;\varphi_{1,{i + 1},1}} + {i_{{c\; 7},i,1}\sin\;\varphi_{7,i,1}} - {i_{{c\; 3},{i + 1},1}\sin\;\varphi_{3,{i + 1},1}}} \right)} - {L_{4,i,1}\left( {{\sum\limits_{k = 1}^{i}{\sin\;\varphi_{1,k,1}}} + {\sum\limits_{k = 1}^{i}{\sin\;\varphi_{2,k,1}}}} \right)}}} & {{Equation}\mspace{14mu}(22)}\end{matrix}$(L _(3a,i,1) +L _(3b,i,1)){dot over(φ)}_(3,i,1)=φ_(2,i,1)−φ_(1,i,1)−φ_(3,i,1)−(L _(3a,i,1) +L _(3b,i,1))i_(c3,i,1) sin φ_(3,i,1)  Equation (23)

$\begin{matrix}{{{L_{{5a},i,1}\left( {{\overset{.}{\varphi}}_{4,i,1} + {\overset{.}{\varphi}}_{6,i,1} - {\overset{.}{\varphi}}_{8,{1 - i},1}} \right)} - {L_{{5b},i,1}\left( {{\overset{.}{\varphi}}_{8,i,1} + {\overset{.}{\varphi}}_{5,i,1} - {\overset{.}{\varphi}}_{6,i,j}} \right)} - {L_{1,i,1}\left( {{\sum\limits_{k = 1}^{i}{\overset{.}{\varphi}}_{1,k,1}} + {\sum\limits_{k = 1}^{i - 1}{\overset{.}{\varphi}}_{2,k,1}} - {\overset{.}{\varphi}}_{3,i,1} - {\sum\limits_{k = 1}^{i}{\overset{.}{\varphi}}_{4,k,1}} - {\sum\limits_{k = 1}^{i - 1}{\overset{.}{\varphi}}_{5,k,1}} - {\overset{.}{\varphi}}_{6,i,1}} \right)}} = {\varphi_{5,i,1} - \varphi_{4,i,1} + {2\pi\; x_{e}a_{2,i,1}} - {L_{{2a},i,1}\left( {{\sin\;\varphi_{4,i,1}} + {i_{{c\; 6},i,1}\sin\;\varphi_{6,i,1}} - {i_{{c\; 8},{i - 1},1}\sin\;\varphi_{8,{i - 1},1}}} \right)} + {L_{{5b},i,1}\left( {{\sin\;\varphi_{5,i,1}} + {i_{{c\; 8},i,1}\sin\;\varphi_{8,i,1}} - {i_{{c\; 6},i,1}\sin\;\varphi_{6,i,1}}} \right)} + {L_{1,i,1}\left( {{\sum\limits_{k = 1}^{i}{\sin\;\varphi_{1,k,1}}} + {\sum\limits_{k = 1}^{i - 1}{\sin\;\varphi_{2,k,1}}} - {i_{{c\; 3},i,1}\sin\;\varphi_{3,i,1}} - {\sum\limits_{k = 1}^{i}{\sin\;\varphi_{4,k,1}}} - {\sum\limits_{k = 1}^{i - 1}{\sin\;\varphi_{5,k,1}}} - {i_{{c\; 6},\; i,1}\sin\;\varphi_{6,i,1}}} \right)}}} & {{Equation}\mspace{14mu}(24)}\end{matrix}$

$\begin{matrix}{{{L_{{5b},i,1}\left( {{\overset{.}{\varphi}}_{5,i,1} + {\overset{.}{\varphi}}_{8,i,1} - {\overset{.}{\varphi}}_{6,i,1}} \right)} - {L_{{5a},{i + 1},1}\left( {{\overset{.}{\varphi}}_{4,{i + 1},1} + {\overset{.}{\varphi}}_{6,{i + 1},1} - {\overset{.}{\varphi}}_{8,i,1}} \right)} + {L_{4,i,2}\left( {{\sum\limits_{k = 1}^{i}{\overset{.}{\varphi}}_{4,k,1}} + {\sum\limits_{k = 1}^{i}{\overset{.}{\varphi}}_{5,k,1}} - {\sum\limits_{k = 1}^{i}{\overset{.}{\varphi}}_{1,k,2}} - {\sum\limits_{k = 1}^{i}{\overset{.}{\varphi}}_{2,k,2}}} \right)}} = {\varphi_{4,{i + 1},1} - \varphi_{5,i,1} + {2\pi\; x_{e}a_{4,i,1}} - {L_{{5b},i,1}\left( {{\sin\;\varphi_{5,i,1}} + {i_{{c\; 8},i,1}\sin\;\varphi_{8,i,1}} - {i_{{c\; 6},i,1}\sin\;\varphi_{6,i,1}}} \right)} + {L_{{5a},{i + 1},1}\left( {{\sin\;\varphi_{4,{i + 1},1}} + {i_{{c\; 6},{i + 1},1}\sin\;\varphi_{6,{i + 1},1}} - {i_{{c\; 8},i,1}\sin\;\varphi_{8,i,1}}} \right)} - {L_{4,i,2}\left( {{\sum\limits_{k = 1}^{i}{\sin\;\varphi_{4,k,1}}} + {\sum\limits_{k = 1}^{i}{\sin\;\varphi_{5,k,1}}} - {\sum\limits_{k = 1}^{i}{\sin\;\varphi_{1,k,2}}} - {\sum\limits_{k = 1}^{i}{\sin\;\varphi_{2,k,2}}}} \right)}}} & {{Equation}\mspace{14mu}(25)}\end{matrix}$(L _(6a,i,1) +L _(6b,i,1)){dot over(φ)}_(6,i,1)=φ_(4,i,1)−φ_(5,i,1)−φ_(6,i,1)−(L _(6a,i,1) +L _(6b,i,1))i_(c3,i,1) sin φ_(3,i,1)  Equation (26)(L _(7a,i,1) +L _(7b,i,1)){dot over (φ)}_(7,i,1) +L _(2a,i+1,1)({dotover (φ)}_(1,i+1,1)+{dot over (φ)}_(7,i,1)−{dot over (φ)}_(3,i+1,1))−L_(2b,i,1)({dot over (φ)}_(2,i,1)+{dot over (φ)}_(3,i,1)−{dot over(φ)}_(7,i,1))=−φ_(7,i,1)−(L _(7a,i,1) +L _(7b,i,1))i _(c7,i,1) sinφ_(7,i,1) −L _(2a,i+1,1)(sin φ_(1,i+1,1) +i _(c7,i,1) sin φ_(7,i,1) −i_(c3,i+1,1) sin φ_(3,i+1,1))+L _(2b,i,1)(sin φ_(2,i,1) +i _(c3,i,1) sinφ_(3,i,1) −i _(c7,i,1) sin φ_(7,i,1))  Equation (27)(L _(8a,i,1) +L _(8b,i,1)){dot over (φ)}_(8,i,1) +L _(5b,i,1)({dot over(φ)}_(5,i,1)+{dot over (φ)}_(8,i,1)−{dot over (φ)}_(6,i,1))−L_(5a,i+1,1)({dot over (φ)}_(4,i+1,1)+{dot over (φ)}_(6,i+1,1)−{dot over(φ)}_(8,i,1))=−φ_(8,i,1)−(L _(8a,i,1) +L _(8b,i,1))i _(c8,i,1) sinφ_(7,i,1) −L _(5b,i,1)(sin φ_(5,i,1) +i _(c8,i,1) sin φ_(8,i,1) −i_(c6,i,1) sin φ_(6,i,1))+L _(5a,i+1,1)(sin φ_(4,i+1,1) +i _(c6,i+1,1)sin φ_(6,i+1,1) −i _(c8,i,1) sin φ_(8,i,1))  Equation (28)

In similar fashion, the phases for the bottom row cells in array 70,i.e., cells 40 _(j,i) for j=M and i=2, . . . , N-1 (8 equations):

$\begin{matrix}{{{L_{{2a},i,M}\left( {{\overset{.}{\varphi}\;}_{1,i,M} + {\overset{.}{\varphi}\;}_{7,{i - 1},M} - {\overset{.}{\varphi}\;}_{3,i,M}} \right)} - {L_{{2b},i,M}\left( {{\overset{.}{\varphi}\;}_{2,i,M} + {\overset{.}{\varphi}\;}_{3,i,M} - {\overset{.}{\varphi}\;}_{7,i,M}} \right)} + {L_{1,i,M}\left( {{\sum\limits_{k = 1}^{i}{\overset{.}{\varphi}}_{1,k,M}} + {\sum\limits_{k = 1}^{i - 1}{\overset{.}{\varphi}}_{2,k,M}} - {\overset{.}{\varphi}}_{3,i,M} - {\sum\limits_{k = 1}^{i}{\overset{.}{\varphi}\;}_{4,k,M}} - {\sum\limits_{k = 1}^{i - 1}{\overset{.}{\varphi}\;}_{5,k,M}} - {\overset{.}{\varphi}\;}_{6,i,M}} \right)}} = {\varphi_{2,i,M} - \varphi_{1,i,M} + {2\pi\; x_{e}a_{1,i,M}} - {L_{{2a},i,M}\left( {{\sin\;\varphi_{1,i,M}} + {i_{{c\; 7},{i - 1},M}\sin\;\varphi_{7,{i - 1},M}} - {i_{{c\; 3},i,M}\sin\;\varphi_{3,i,M}}} \right)} + {L_{{2b},i,M}\left( {{\sin\;\varphi_{2,i,M}} + {i_{{c\; 3},i,M}\sin\;\varphi_{3,i,M}} - {i_{{c\; 7},i,M}\sin\;\varphi_{7,i,M}}} \right)} - {L_{1,i,M}\left( {{\sum\limits_{k = 1}^{i}{\sin\;\varphi_{1,k,M}}} + {\sum\limits_{k = 1}^{i - 1}{\sin\;\varphi_{2,k,M}}} - {i_{{c\; 3},i,M}\sin\;\varphi_{3,i,M}} - {\sum\limits_{k = 1}^{i}{\sin\;\varphi_{4,k,M}}} - {\sum\limits_{k = 1}^{i - 1}{\sin\;\varphi_{5,k,M}}} - {i_{{c\; 6},i,M}\sin\;\varphi_{6,i,M}}} \right)}}} & {{Equation}\mspace{14mu}(29)}\end{matrix}$

$\begin{matrix}{{{L_{{2b},i,M}\left( {{\overset{.}{\varphi}}_{2,i,M} + {\overset{.}{\varphi}}_{3,i,M} - {\overset{.}{\varphi}}_{7,i,M}} \right)} - {L_{{2a},{i + 1},M}\left( {{\overset{.}{\varphi}}_{1,{i + 1},M} + {\overset{.}{\varphi}}_{7,i,M} - {\overset{.}{\varphi}}_{3,{i + 1},M}} \right)} - {L_{4,i,M}\left( {{\sum\limits_{k = 1}^{i}{\overset{.}{\varphi}}_{4,k,{M - 1}}} + {\sum\limits_{k = 1}^{i}{\overset{.}{\varphi}}_{5,k,{M - 1}}} - {\sum\limits_{k = 1}^{i}{\overset{.}{\varphi}}_{1,k,M}} - {\sum\limits_{k = 1}^{i}{\overset{.}{\varphi}}_{2,k,M}}} \right)}} = {\varphi_{1,{i + 1},M} - \varphi_{2,i,M} + {2\pi\; x_{e}a_{3,i,M}} - {L_{{2b},i,M}\left( {{\sin\;\varphi_{2,i,M}} + {i_{{c\; 3},i,M}\sin\;\varphi_{3,i,M}} - {i_{{c\; 7},i,M}\sin\;\varphi_{7,i,M}}} \right)} + {L_{{2a},{i + 1},M_{i}}\left( {{\sin\;\varphi_{1,{i + 1},M}} + {i_{{c\; 7},i,M}\sin\;\varphi_{7,i,M}} - {i_{{c\; 3},{i + 1},M}\sin\;\varphi_{3,{i + 1},M}}} \right)} + {L_{4,i,M}\left( {{\sum\limits_{k = 1}^{i}{\sin\;\varphi_{4,k,{M - 1}}}} + {\sum\limits_{k = 1}^{i}{\sin\;\varphi_{5,k,{M - 1}}}} - {\sum\limits_{k = 1}^{i}{\sin\;\varphi_{i,k,M}}} - {\sum\limits_{k = 1}^{i}{\sin\;\varphi_{2,k,M}}}} \right)}}} & {{Equation}\mspace{14mu}(30)}\end{matrix}$(L _(3a,i,M) +L _(3b,i,M)){dot over(φ)}_(3,i,M)=φ_(2,i,M)−φ_(1,i,M)−φ_(3,i,M)−(L _(3a,i,M) +L _(3b,i,M))i_(c3,i,M) sin φ_(3,i,M)  Equation (31)

$\begin{matrix}{{{L_{{5a},i,M}\left( {{\overset{.}{\varphi}}_{4,i,M} + {\overset{.}{\varphi}}_{6,i,M} - {\overset{.}{\varphi}}_{8,{i - 1},M}} \right)} - {L_{{5b},i,M}\left( {{\overset{.}{\varphi}}_{8,i,M} + {\overset{.}{\varphi}}_{5,i,M} - {\overset{.}{\varphi}}_{6,i,M}} \right)} - {L_{1,i,M}\left( {{\sum\limits_{k = 1}^{i}{\overset{.}{\varphi}}_{1,k,M}} + {\sum\limits_{k = 1}^{i - 1}{\overset{.}{\varphi}}_{2,k,M}} - {\overset{.}{\varphi}}_{3,i,M} - {\sum\limits_{k = 1}^{i}{\overset{.}{\varphi}}_{4,k,M}} - {\sum\limits_{k = 1}^{i - 1}{\overset{.}{\varphi}}_{5,k,M}} - {\overset{.}{\varphi}}_{6,i,M}} \right)}} = {\varphi_{5,i,M} - \varphi_{4,i,M} + {2\pi\; x_{e}a_{2,i,M}} - {L_{{2a},i,M}\left( {{\sin\;\varphi_{4,i,M}} + {i_{{c\; 6},i,M}\sin\;\varphi_{6,i,M}} - {i_{{c\; 8},{i - 1},M}\sin\;\varphi_{8,{i - 1},M}}} \right)} + {L_{{5b},i,M}\left( {{\sin\;\varphi_{5,i,M}} + {i_{{c\; 8},i,M}\sin\;\varphi_{8,i,M}} - {i_{{c\; 6},i,M}\sin\;\varphi_{6,i,M}}} \right)} + {L_{1,i,M}\left( {{\sum\limits_{k = 1}^{i}{\sin\;\varphi_{1,k,M}}} + {\sum\limits_{k = 1}^{i - 1}{\sin\;\varphi_{2,k,M}}} - {i_{{c\; 3},i,M}\sin\;\varphi_{3,i,M}} - {\sum\limits_{k = 1}^{i}{\sin\;\varphi_{4,k,M}}} - {\sum\limits_{k = 1}^{i - 1}{\sin\;\varphi_{5,k,M}}} - {i_{{c\; 6},i,M}\sin\;\varphi_{6,i,M}}} \right)}}} & {{Equation}\mspace{14mu}(32)}\end{matrix}$

$\begin{matrix}{{{L_{{5b},i,M}\left( {{\overset{.}{\varphi}}_{5,i,M} + {\overset{.}{\varphi}}_{8,i,M} - {\overset{.}{\varphi}}_{6,i,M}} \right)} - {L_{{5a},{i + 1},M}\left( {{\overset{.}{\varphi}}_{4,{i + 1},M} + {\overset{.}{\varphi}}_{6,{i + 1},M} - {\overset{.}{\varphi}}_{8,i,M}} \right)} + {L_{4,i,{M + 1}}\left( {{\sum\limits_{k = 1}^{i}{\overset{.}{\varphi}}_{4,k,M}} + {\sum\limits_{k = 1}^{i}{\overset{.}{\varphi}}_{5,k,M}}} \right)}} = {{i \times i_{b}L_{4,i,{M + 1}}} + \varphi_{4,{i + 1},M} - \varphi_{5,i,M} + {2\pi\; x_{e}a_{4,i,M}} - {L_{{5b},i,M}\left( {{\sin\;\varphi_{5,i,M}} + {i_{{c\; 8},i,M}\sin\;\varphi_{8,i,M}} - {i_{{c\; 6},i,M}\sin\;\varphi_{6,i,M}}} \right)} + {L_{{5a},{i + 1},M}\left( {{\sin\;\varphi_{4,{i + 1},M}} + {i_{{c\; 6},{i + 1},M}\sin\;\varphi_{6,{i + 1},M}} - {i_{{c\; 8},i,M}\sin\;\varphi_{8,i,M}}} \right)} - {L_{4,i,{M + 1}}\left( {{\sum\limits_{k = 1}^{i}{\sin\;\varphi_{4,k,M}}} + {\sum\limits_{k = 1}^{i}{\sin\;\varphi_{5,k,M}}}} \right)}}} & {{Equation}\mspace{14mu}(33)}\end{matrix}$(L _(6a,i,M) +L _(6b,i,M)){dot over(φ)}_(6,i,M)=φ_(4,i,M)−φ_(5,i,M)−φ_(6,i,M)−(L _(6a,i,M) +L _(6b,i,M))i_(c3,i,M) sin φ_(3,i,M)  Equation (34)(L _(7a,i,M) +L _(7b,i,M)){dot over (φ)}_(7,i,M) +L _(2a,i+1,M)({dotover (φ)}_(1,i+1,M)+{dot over (φ)}_(7,i,M)−{dot over (φ)}_(3,i+1,M))−L_(2b,i,M)({dot over (φ)}_(2,i,M)+{dot over (φ)}_(3,i,M)−{dot over(φ)}_(7,i,M))=−φ_(7,i,M)−(L _(7a,i,M) +L _(7b,i,M))i _(c7,i,M) sinφ_(7,i,M) −L _(2a,i+1,M)(sin φ_(1,i+1,M) +i _(c7,i,M) sin φ_(7,i,M) −i_(c3,i+1,M) sin φ_(3,i+1,M))+L _(2b,i,M)(sin φ_(2,i,M) +i _(c3,i,M) sinφ_(3,i,M) −i _(c7,i,M) sin φ_(7,i,M))  Equation (35)(L _(8a,i,M) +L _(8b,i,M)){dot over (φ)}_(8,i,M) +L _(5b,i,M)({dot over(φ)}_(5,i,M)+{dot over (φ)}_(8,i,M)−{dot over (φ)}_(6,i,M))−L_(5a,i+1,M)({dot over (φ)}_(4,i+1,M)+{dot over (φ)}_(6,i+1,M)−{dot over(φ)}_(8,i,M))=−φ_(8,i,M)−(L _(8a,i,M) +L _(8b,i,M))i _(c8,i,M) sinφ_(7,i,M) −L _(5b,i,M)(sin φ_(5,i,M) +i _(c8,i,M) sin φ_(8,i,M) −i_(c6,i,M) sin φ_(6,i,M))+L _(5a,i+1,M)(sin φ_(4,i+1,M) +i _(c6,i+1,M)sin φ_(6,i+1,M) −i _(c8,i,M) sin φ_(8,i,M))  Equation (36)

For the cells 40 on the left side cells of 2D SQIF array 70, i.e. forcells 40 _(j,i) for j=2, . . . , M-1 and i=1 (8 equations):L _(2a,1,j)({dot over (φ)}_(1,1,j)−{dot over (φ)}_(3,1,j))−L_(2b,1,j)({dot over (φ)}_(2,1,j)+{dot over (φ)}_(3,1,j)−{dot over(φ)}_(7,1,j))+L _(1,1,j)({dot over (φ)}_(1,1,j)−{dot over(φ)}_(3,1,j)−{dot over (φ)}_(4,1,j)−{dot over(φ)}_(6,1,j))=φ_(2,1,j)−φ_(1,1,j)+2πx _(e) a _(1,1,j) −L _(2a,1,j)(sinφ_(1,1,j) −i _(c3,1,j) sin φ_(3,1,j))+L _(2b,1,j)(sin φ_(2,1,j) +i_(c3,1,j) sin φ_(3,1,j) −i _(c7,1,j) sin φ_(7,1,j))−L _(1,1,j)(sinφ_(1,1,j) −i _(c3,1,j) sin φ_(3,1,j)−sin φ_(4,1,j) −i _(c6,1,j) sinφ_(6,1,j))  Equation (37)L _(2b,1,j)({dot over (φ)}_(2,1,j)+{dot over (φ)}_(3,1,j)−{dot over(φ)}_(7,1,j))−L _(2a,2,j)({dot over (φ)}_(1,2,j)+{dot over(φ)}_(7,1,j)−{dot over (φ)}_(3,2,j))−L _(4,1,j)({dot over(φ)}_(4,1,j−1)+{dot over (φ)}_(5,1,j−1)−{dot over (φ)}_(1,1,j)−{dot over(φ)}_(2,1,j))=φ_(1,2,j)−φ_(2,1,j)+2πx _(e) a _(3,1,j) −L _(2b,1,j)(sinφ_(2,1,j) +i _(c3,1,j) sin φ_(3,1,j) −i _(c7,1,j) sin φ_(7,1,j))+L_(2a,2,j)(sin φ_(1,2,j) +i _(c7,1,j) sin φ_(7,1,j) −i _(c3,2,j) sinφ_(3,2,j))+L _(4,1,j)(sin φ_(4,1,j−1)+sin φ_(5,1,j−1)−sin φ_(1,1,j)−sinφ_(2,1,j))  Equation (38)(L _(3a,1,j) +L _(3b,1,j)){dot over(φ)}_(3,i,j)=φ_(2,1,j)−φ_(1,1,j)−φ_(3,1,j)−(L _(3a,1,j) +L _(3b,1,j))i_(c3,1,j) sin φ_(3,1,j)  Equation (39)L _(5a,1,j)({dot over (φ)}_(4,1,j)+{dot over (φ)}_(6,1,j))−L_(5b,1,j)({dot over (φ)}_(8,1,j)+{dot over (φ)}_(5,1,j)−{dot over(φ)}_(6,1,j))−L _(1,1,j)({dot over (φ)}_(1,1,j)−{dot over(φ)}_(3,1,j)−{dot over (φ)}_(4,1,j)−{dot over(φ)}_(6,1,j))=φ_(5,1,j)−φ_(4,1,j)+2πx _(e) a _(2,1,j) −L _(2a,1,j)(sinφ_(4,1,j) +i _(c6,1,j) sin φ_(6,1,j))+L _(5b,1,j)(sin φ_(5,1,j) +i_(c8,1,j) sin φ_(8,1,j) −i _(c6,1,j) sin φ_(6,1,j))+L _(1,1,j)(sinφ_(1,1,j) −i _(c3,1,j)+sin φ_(3,1,j)−sin φ_(4,1,j) −i _(c6,1,j) sinφ_(6,1,j))  Equation (40)L _(5b,1,j)({dot over (φ)}_(5,1,j)+{dot over (φ)}_(8,1,j)−{dot over(φ)}_(6,1,j))−L _(5a,2,j)({dot over (φ)}_(4,2,j)+{dot over(φ)}_(6,2,j)−{dot over (φ)}_(8,1,j))+L _(4,1,j+1)({dot over(φ)}_(4,1,j)+{dot over (φ)}_(5,1,j)−{dot over (φ)}_(1,1,j+1)−{dot over(φ)}_(2,1,j+1))=φ_(4,2,j)−φ_(5,1,j)+2πx _(e) a _(4,1,j) −L _(5b,1,j)(sinφ_(5,1,j) +i _(c8,1,j) sin φ_(8,1,j) −i _(c6,1,j) sin φ_(6,1,j))+L_(5a,2,j)(sin φ_(4,2,j) +i _(c6,2,j) sin φ_(6,2,j) −i _(c8,1,j) sinφ_(8,1,j))−L _(4,1,j+1)(sin φ_(4,1,j)+sin φ_(5,1,j)−sin φ_(1,1,j+1)−sinφ_(2,1,j+1))  Equation (41)(L _(6a,1,j) +L _(6b,1,j)){dot over(φ)}_(6,1,j)=φ_(4,1,j)−φ_(5,1,j)−φ_(6,1,j)−(L _(6a,1,j) +L _(6b,1,j))i_(c3,1,j) sin φ_(3,1,j)  Equation (42)(L _(7a,1,j) +L _(7b,1,j)){dot over (φ)}_(7,1,j) +L _(2a,2,j)({dot over(φ)}_(1,2,j)+{dot over (φ)}_(7,1,j)−{dot over (φ)}_(3,2,j))−L_(2b,1,j)({dot over (φ)}_(2,1,j)+{dot over (φ)}_(3,1,j)−{dot over(φ)}_(7,1,j))=−φ_(7,1,j)−(L _(7a,1,j) +L _(7b,1,j))i _(c7,1,j) sinφ_(7,1,j) −L _(2a,2,j)(sin φ_(1,2,j) +i _(c7,1,j) sin φ_(7,1,j) −i_(c3,2,j) sin φ_(3,2,j))+L _(2b,1,j)(sin φ_(2,1,j) +i _(c3,1,j) sinφ_(3,1,j) −i _(c7,1,j) sin φ_(7,1,j))  Equation (43)(L _(8a,1,j) +L _(8b,1,j)){dot over (φ)}_(8,1,j) +L _(5b,1,j)({dot over(φ)}_(5,1,j)+{dot over (φ)}_(8,1,j)−{dot over (φ)}_(6,1,j))−L_(5a,2,j)({dot over (φ)}_(4,2,j)+{dot over (φ)}_(6,2,j)−{dot over(φ)}_(8,1,j))=−φ_(8,1,j)−(L _(8a,1,j) +L _(8b,1,j))i _(c8,1,j) sinφ_(7,1,j) −L _(5b,1,j)(sin φ_(5,1,j) +i _(c8,1,j) sin φ_(8,1,j) −i_(c6,1,j) sin φ_(6,1,j))+L _(5a,2,j)(sin φ_(4,2,j) +i _(c6,2,j) sinφ_(6,2,j) −i _(c8,1,j) sin φ_(8,1,j))  Equation (44)

To model the boundary conditions at the corners of the 2D SQIF array 70,and startin with top left-hand corner cell 40 _(j,i) for j=1 and i=1 (8equations):L _(2a,1,1)({dot over (φ)}_(1,1,1)−{dot over (φ)}_(3,1,1))−L_(2b,1,1)({dot over (φ)}_(2,1,1)+{dot over (φ)}_(3,1,1)−{dot over(φ)}_(7,1,1))+L _(1,1,1)({dot over (φ)}_(1,1,1)−{dot over(φ)}_(3,1,1)−{dot over (φ)}_(4,1,1)−{dot over(φ)}_(6,1,1))=φ_(2,1,1)−φ_(1,1,1)+2πx _(e) a _(1,1,1) −L _(2a,1,1)(sinφ_(1,1,1) −i _(c3,1,1) sin φ_(3,1,1))+L _(2b,1,1)(sin φ_(2,1,1) +i_(c3,1,1) sin φ_(3,1,1) −i _(c7,1,1) sin φ_(7,1,1))−L _(1,1,1)(sinφ_(1,1,1) −i _(c3,1,1) sin φ_(3,1,1)−sin φ_(4,1,1) −i _(c6,1,1) sinφ_(6,1,1))  Equation (45)L _(2b,1,1)({dot over (φ)}_(2,1,1)+{dot over (φ)}_(3,1,1)−{dot over(φ)}_(7,1,1))−L _(2a,2,1)({dot over (φ)}_(1,2,1)+{dot over(φ)}_(7,1,1)−{dot over (φ)}_(3,2,1))+L _(4,1,1)({dot over(φ)}_(1,1,1)+{dot over (φ)}_(2,1,1))=L _(4,1,1) i_(b)+φ_(1,2,1)−φ_(2,1,1)+2πx _(e) a _(3,1,1) −L _(2b,1,1)(sin φ_(2,1,1)+i _(c3,1,1) sin φ_(3,1,1) −i _(c7,1,1) sin φ_(7,1,1))+L _(2a,2,1)(sinφ_(1,2,1) +i _(c7,1,1) sin φ_(7,1,1) −i _(c3,2,1) sin φ_(3,2,1))−L_(4,1,1)(sin φ_(1,1,1)+sin φ_(2,1,1))  Equation (46)(L _(3a,1,1) +L _(3b,1,1)){dot over(φ)}_(3,1,1)=φ_(2,1,1)−φ_(1,1,)−φ_(3,1,1)−(L _(3a,1,1) +L _(3b,1,1))i_(c3,1,1) sin φ_(3,1,1)  Equation (47)L _(5a,1,1)({dot over (φ)}_(4,1,1)+{dot over (φ)}_(6,1,1))−L_(5b,1,1)({dot over (φ)}_(8,1,1)+{dot over (φ)}_(5,1,1)−{dot over(φ)}_(6,1,1))−L _(1,1,1)({dot over (φ)}_(1,1,1)−{dot over(φ)}_(3,1,1)−{dot over (φ)}_(4,1,1)−{dot over(φ)}_(6,1,1))=φ_(5,1,1)−φ_(4,1,1)+2πx _(e) a _(2,1,1) −L _(2a,1,1)(sinφ_(4,1,1) +i _(c6,1,1) sin φ_(6,1,1))+L _(5b,1,1)(sin φ_(5,1,1) +i_(c8,1,1) sin φ_(8,1,1) −i _(c6,1,1) sin φ_(6,1,1))+L _(1,1,1)(sinφ_(1,1,1) −i _(c3,1,1) sin φ_(3,1,1)−sin φ_(4,1,1) −i _(c6,1,1) sinφ_(6,1,))  Equation (48)L _(5b,1,1)({dot over (φ)}_(5,1,1)+{dot over (φ)}_(8,1,1)−{dot over(φ)}_(6,1,1))−L _(5a,2,1)({dot over (φ)}_(4,2,1)+{dot over(φ)}_(6,2,1)−{dot over (φ)}_(8,1,1))+L _(4,1,2)({dot over(φ)}_(4,1,1)+{dot over (φ)}_(5,1,1)−{dot over (φ)}_(1,1,2)−{dot over(φ)}_(2,1,2))=φ_(4,2,1)−φ_(5,1,1)+2πx _(e) a _(4,1,1) −L _(5b,1,1)(sinφ_(5,1,1) +i _(c8,1,1) sin φ_(8,1,1) −i _(c6,1,1) sin φ_(6,1,1))+L_(5a,2,1)(sin φ_(4,2,1) +i _(c6,2,1) sin φ_(6,2,1) −i _(c8,1,1) sinφ_(8,1,1))−L _(4,1,2)(sin φ_(4,1,1)+sin φ_(5,1,1)−sin φ_(1,1,2)−sinφ_(2,1,2))  Equation (49)(L _(6a,1,1) +L _(6b,1,1)){dot over(φ)}_(6,1,1)=φ_(4,1,1)−φ_(5,1,1)−φ_(6,1,1)−(L _(6a,1,1) +L _(6b,1,1))i_(c3,1,1) sin φ_(3,1,1)  Equation (50)(L _(7a,1,1) +L _(7b,1,1)){dot over (φ)}_(7,1,1) +L _(2a,2,1)({dot over(φ)}_(1,2,1)+{dot over (φ)}_(7,1,1)−{dot over (φ)}_(3,2,1))−L_(2b,1,1)({dot over (φ)}_(2,1,1)+{dot over (φ)}_(3,1,1)−{dot over(φ)}_(7,1,1))=−φ_(7,1,1)−(L _(7a,1,1) +L _(7b,1,1))i _(c7,1,1) sinφ_(7,1,1) −L _(2a,2,1)(sin φ_(1,2,1) +i _(c7,1,1) sin φ_(7,1,1) −i_(c3,2,1) sin φ_(3,2,1))+L _(2b,1,1)(sin φ_(2,1,1) +i _(c3,1,1) sinφ_(3,1,1) −i _(c7,1,1) sin φ_(7,1,1))  Equation (51)(L _(8a,1,1) +L _(8b,1,1)){dot over (φ)}_(8,1,1) +L _(5b,1,1)({dot over(φ)}_(5,1,1)+{dot over (φ)}_(8,1,1)−{dot over (φ)}_(6,1,1))−L_(5a,2,1)({dot over (φ)}_(4,2,1)+{dot over (φ)}_(6,2,1)−{dot over(φ)}_(8,1,1))=−φ_(8,1,1)−(L _(8a,1,1) +L _(8b,1,1))i _(c8,1,1) sinφ_(7,1,1) −L _(5b,1,1)(sin φ_(5,1,1) +i _(c8,1,1) sin φ_(8,1,1) −i_(c6,1,1) sin φ_(6,1,1))+L _(5a,2,1)(sin φ_(4,2,1) +i _(c6,2,1) sinφ_(6,2,1) −i _(c8,1,1) sin φ_(8,1,1))  Equation (52)

For the bottom left-hand corner cell 40 _(j,i) for j=M and i=1:L _(2a,1,M)({dot over (φ)}_(1,1,M)−{dot over (φ)}_(3,1,M))−L_(2b,1,M)({dot over (φ)}_(2,1,M)+{dot over (φ)}_(3,1,M)−{dot over(φ)}_(7,1,M))+L _(1,1,M)({dot over (φ)}_(1,1,M)−{dot over(φ)}_(3,1,M)−{dot over (φ)}_(4,1,M)−{dot over(φ)}_(6,1,M))=φ_(2,1,M)−φ_(1,1,M)+2πx _(e) a _(1,1,M) −L _(2a,1,M)(sinφ_(1,1,M) −i _(c3,1,M) sin φ_(3,1,M))+L _(2b,1,M)(sin φ_(2,1,M) +i_(c3,1,M) sin φ_(3,1,M) −i _(c7,1,M) sin φ_(7,1,M))−L _(1,1,M)(sinφ_(1,1,M) −i _(c3,1,M) sin φ_(3,1,M)−sin φ_(4,1,M) −i _(c6,1,M) sinφ_(6,1,M))  Equation (53)L _(2b,1,M)({dot over (φ)}_(2,1,M)+{dot over (φ)}_(3,1,M)−{dot over(φ)}_(7,1,M))−L _(2a,2,M)({dot over (φ)}_(1,2,M)+{dot over(φ)}_(7,1,M)−{dot over (φ)}_(3,2,M))−L _(4,1,M+1)({dot over(φ)}_(4,1,M−1)+{dot over (φ)}_(5,1,M−1)−{dot over (φ)}_(1,1,M)−{dot over(φ)}_(2,1,M))=φ_(1,2,M)−φ_(2,1,M)+2πx _(e) a _(3,1,M) −L _(2b,1,M)(sinφ_(2,1,M) +i _(c3,1,M) sin φ_(3,1,M) −i _(c7,1,M) sin φ_(7,1,M))+L_(2a,2,M)(sin φ_(1,2,M) +i _(c7,1,M) sin φ_(7,1,M) −i _(c3,2,M) sinφ_(3,2,M))+L _(4,1,M)(sin φ_(4,1,M−1)+sin φ_(5,1,M−1)−sin φ_(1,1,M)−sinφ_(2,1,M))  Equation (54)(L _(3a,1,M) +L _(3b,1,M)){dot over(φ)}_(3,1,M)=φ_(2,1,M)−φ_(1,1,M)−φ_(3,1,M)−(L _(3a,1,M) +L _(3b,1,M))i_(c3,1,M) sin φ_(3,1,M)  Equation (55)L _(5a,1,M)({dot over (φ)}_(4,1,M)+{dot over (φ)}_(6,1,M))−L_(5b,1,M)({dot over (φ)}_(8,1,M)+{dot over (φ)}_(5,1,M)−{dot over(φ)}_(6,1,M))−L _(1,1,M)({dot over (φ)}_(1,1,M)−{dot over(φ)}_(3,1,M)−{dot over (φ)}_(4,1,M)−{dot over(φ)}_(6,1,M))=φ_(5,1,M)−φ_(4,1,M)+2πx _(e) a _(2,1,M) −L _(2a,1,M)(sinφ_(4,1,M) +i _(c6,1,M) sin φ_(8,1,M))+L _(5b,1,M)(sin φ_(5,1,M) +i_(c8,1,M) sin φ_(8,1,M) −i _(c6,1,M) sin φ_(6,1,M))+L _(1,1,M)(sinφ_(1,1,M) −i _(c3,1,M) sin φ_(3,1,M)−sin φ_(4,1,M) −i c6,1,M sinφ_(6,1,M))  Equation (56)L _(5b,1,M)({dot over (φ)}_(5,1,M)+{dot over (φ)}_(8,1,M)−{dot over(φ)}_(6,1,M))−L _(5a,2,M)({dot over (φ)}_(4,2,M)+{dot over(φ)}_(6,2,M)−{dot over (φ)}_(8,1,M))+L _(4,1,M+1)({dot over(φ)}_(4,1,M)+{dot over (φ)}_(5,1,M))=φ_(4,2,M)−φ_(5,1,M)+2πx _(e) a_(4,1,M) −L _(5b,1,M)(sin φ_(5,1,M) +i _(c8,1,M) sin φ_(8,1,M) −i_(c6,1,M) sin φ_(6,1,M))+L _(5a,2,M)(sin φ_(4,2,M) +i _(c6,2,M) sinφ_(6,2,M) −i _(c8,1,M) sin φ_(8,1,M))−L _(4,1,M+1)(sin φ_(4,1,M)+sinφ_(5,1,M))  Equation (57)(L _(6a,1,M) +L _(6b,1,M)){dot over(φ)}_(6,1,M)=φ_(4,1,M)−φ_(5,1,M)−φ_(6,1,M)−(L _(6a,1,M) +L _(6b,1,M))i_(c3,1,M) sin φ_(3,1,M)  Equation (58)(L _(7a,1,M) +L _(7b,1,M)){dot over (φ)}_(7,1,M) +L _(2a,2,M)({dot over(φ)}_(1,2,M)+{dot over (φ)}_(7,1,M)−{dot over (φ)}_(3,2,M))−L_(2b,1,M)({dot over (φ)}_(2,1,M)+{dot over (φ)}_(3,1,M)−{dot over(φ)}_(7,1,M))=−φ_(7,1,M)−(L _(7a,1,M) +L _(7b,1,M))i _(c7,1,M) sinφ_(7,1,M) −L _(2a,2,M)(sin φ_(1,2,M) +i _(c7,1,M) sin φ_(7,1,M) −i_(c3,2,M) sin φ_(3,2,M))+L _(2b,1,M)(sin φ_(2,1,M) +i _(c3,1,M) sinφ_(3,1,M) −i _(c7,1,M) sin φ_(7,1,M))  Equation (59)(L _(8a,1,M) +L _(8b,1,M)){dot over (φ)}_(8,1,M) +L _(5b,1,M)({dot over(φ)}_(5,1,M)+{dot over (φ)}_(8,1,M)−{dot over (φ)}_(6,1,M))−L_(5a,2,M)({dot over (φ)}_(4,2,M)+{dot over (φ)}_(6,2,M)−{dot over(φ)}_(8,1,M))=−φ_(8,1,M)−(L _(8a,1,M) +L _(8b,1,M))i _(c8,1,M) sinφ_(7,1,M) −L _(5b,1,M)(sin φ_(5,1,M) +i _(c8,1,M) sin φ_(8,1,M) −i_(c6,1,M) sin φ_(6,1,M))+L _(5a,2,M)(sin φ_(4,2,M) +i _(c6,2,M) sinφ_(6,2,M) −i _(c8,1,M) sin φ_(8,1,M))  Equation (60)

For the right side cells of 2D SQIF array 70, i.e. for cells 40 _(j,i)for j=2, . . . , M-1 and i=N (6 equations):

$\begin{matrix}{{{L_{{2a},N,j}{\overset{.}{\varphi}}_{1,N,j}} - {\left( {L_{1,N,j} + L_{{2b},N,j}} \right)\left( {{\sum\limits_{k = 1}^{N}{\overset{.}{\varphi}}_{4,k,{j - 1}}} + {\sum\limits_{k = 1}^{N}{\overset{.}{\varphi}}_{5,k,{j - 1}}} - {\sum\limits_{k = 1}^{N}{\overset{.}{\varphi}}_{1,k,j}} - {\sum\limits_{k = 1}^{N - 1}{\overset{.}{\varphi}}_{2,k,j}}} \right)} - {\left( {L_{1,N,j} + L_{{2a},N,j} + L_{{2b},N,j}} \right){\overset{.}{\varphi}}_{3,N,j}} + {L_{1,N,j}{\overset{.}{\varphi}}_{5,N,j}} - {L_{1,N,j}{\overset{.}{\varphi}}_{6,N,j}} + {L_{{2a},N,j}{\overset{.}{\varphi}}_{7,{N - 1},j}}} = {\varphi_{2,N,j} - \varphi_{1,N,j} + {2\pi\; x_{e}a_{1,N,j}} - {L_{{2a},N,j}\sin\;\varphi_{1,N,j}} + {\left( {L_{1,N,j} + L_{{2b},N,j}} \right)\left( {{\sum\limits_{k = 1}^{N}{\sin\;\varphi_{4,k,{j - 1}}}} + {\sum\limits_{k = 1}^{N}{\sin\;\varphi_{5,k,{j - 1}}}} - {\sum\limits_{k = 1}^{N}{\sin\;\varphi_{1,k,j}}} - {\sum\limits_{k = 1}^{N - 1}{\sin\;\varphi_{2,k,j}}}} \right)} + {\left( {L_{1,N,j} + L_{{2a},N,j} + L_{{2b},N,j}} \right)i_{{c\; 3},N,j}\sin\;\varphi_{3,N,j}} - {L_{1,N,j}\sin\;\varphi_{5,N,j}} + {L_{1,N,j}i_{{c\; 6},N,j}\sin\;\varphi_{6,N,j}} - {L_{{2a},N,j}i_{{c\; 7},{N - 1},j}\sin\;\varphi_{7,{N - 1},j}}}} & {{Equation}\mspace{14mu}(61)}\end{matrix}$

$\begin{matrix}{{{L_{{2a},N,j}\left( {{\sum\limits_{k = 1}^{N}{\overset{.}{\varphi}}_{4,k,{j - 1}}} + {\sum\limits_{k = 1}^{N}{\overset{.}{\varphi}}_{5,k,{j - 1}}} - {\sum\limits_{k = 1}^{N - 1}{\overset{.}{\varphi}}_{1,k,j}} - {\sum\limits_{k = 1}^{N}{\overset{.}{\varphi}}_{2,k,j}}} \right)} - {\left( {L_{1,N,j} + L_{{2b},N,j}} \right){\overset{.}{\varphi}}_{2,N,j}} - {\left( {L_{1,N,j} + L_{{2a},N,j} + L_{{2b},N,j}} \right){\overset{.}{\varphi}}_{3,N,j}} + {L_{1,N,j}{\overset{.}{\varphi}}_{5,N,j}} - {L_{1,N,j}{\overset{.}{\varphi}}_{6,N,j}} + {L_{{2a},N,j}{\overset{.}{\varphi}}_{7,{N - 1},j}}} = {\varphi_{2,N,j} - \varphi_{1,N,j} + {2\pi\; x_{e}a_{1,N,j}} - {L_{{2a},N,j}\left( {{\sum\limits_{k = 1}^{N}{\sin\;\varphi_{4,k,{j - 1}}}} + {\sum\limits_{k = 1}^{N}{\sin\;\varphi_{5,k,{j - 1}}}} - {\sum\limits_{k = 1}^{N - 1}{\sin\;\varphi_{1,k,j}}} - {\sum\limits_{k = 1}^{N}{\sin\;\varphi_{2,k,j}}}} \right)} + {\left( {L_{1,N,j} + L_{{2b},N,j}} \right)\sin\;\varphi_{2,N,j}} + {\left( {L_{1,N,j} + L_{{2a},N,j} + L_{{2b},N,j}} \right)i_{{c\; 3},N,j}\sin\;\varphi_{3,\; N,j}} - {L_{1,N,j}\sin\;\varphi_{5,N,j}} + {L_{1,N,j}i_{{c\; 6},N,j}\sin\;\varphi_{6,N,j}} - {L_{{2a},N,j}i_{{c\; 7},{N - 1},j}\sin\;\varphi_{7,{N - 1},j}}}} & {{Equation}\mspace{14mu}(62)}\end{matrix}$(L _(3a,N,j) +L _(3b,N,j)){dot over(φ)}_(3,N,j)=φ_(2,N,j)−φ_(1,N,j)−φ_(3,N,j)−(L _(3a,N,j) +L _(3b,N,j))i_(c3,N,j) sin φ_(3,N,j)  Equation (63)

$\begin{matrix}{{{L_{{5a},N,j}{\overset{.}{\varphi}}_{4,N,j}} - {\left( {L_{1,N,j} + L_{{5b},N,j}} \right)\left( {{\sum\limits_{k = 1}^{N}{\overset{.}{\varphi}}_{1,k,j}} + {\sum\limits_{k = 1}^{N}{\overset{.}{\varphi}}_{2,k,j}} - {\sum\limits_{k = 1}^{N}{\overset{.}{\varphi}}_{4,k,j}} - {\sum\limits_{k = 1}^{N - 1}{\overset{.}{\varphi}}_{5,k,j}}} \right)} + {\left( {L_{1,N,j} + L_{{5a},N,j} + L_{{5b},N,j}} \right){\overset{.}{\varphi}}_{6,N,j}} + {L_{1,N,j}{\overset{.}{\varphi}}_{2,N,j}} + {L_{1,N,j}{\overset{.}{\varphi}}_{3,N,j}} - {L_{{5a},N,j}{\overset{.}{\varphi}}_{8,{N - 1},j}}} = {\varphi_{5,N,j} - \varphi_{4,N,j} + {2\pi\; x_{e}a_{2,N,j}} - {L_{{5a},N,j}\sin\;\varphi_{4,N,j}} + {\left( {L_{1,N,j} + L_{{5b},N,j}} \right)\left( {{\sum\limits_{k = 1}^{N}{\sin\;\varphi_{1,k,j}}} + {\sum\limits_{k = 1}^{N}{\sin\;\varphi_{2,k,j}}} - {\sum\limits_{k = 1}^{N}{\sin\;\varphi_{4,k,j}}} - {\sum\limits_{k = 1}^{N - 1}{\sin\;\varphi_{5,k,j}}}} \right)} - {\left( {L_{1,N,j} + L_{{5a},N,j} + L_{{5b},N,j}} \right)i_{{c\; 6},N,j}\sin\;\varphi_{6,N,j}} - {L_{1,N,j}\sin\;\varphi_{2,N,j}} - {L_{1,N,j}i_{{c\; 3},N,j}\sin\;\varphi_{3,N,j}} + {L_{{5a},N,j}i_{{c\; 8},{N - 1},j}\sin\;\varphi_{8,{N - 1},j}}}} & {{Equation}\mspace{14mu}(64)}\end{matrix}$

$\begin{matrix}{{{L_{{5\; a},N,j}\left( {{\sum\limits_{k = 1}^{N}{\overset{.}{\varphi}}_{1,k,j}} + {\sum\limits_{k = 1}^{N}{\overset{.}{\varphi}}_{2,k,j}} - {\sum\limits_{k = 1}^{N}{\overset{.}{\varphi}}_{4,k,j}} - {\sum\limits_{k = 1}^{N}{\overset{.}{\varphi}}_{5,k,j}}} \right)} - {\left( {L_{1,N,j} + L_{{5b},N,j}} \right){\overset{.}{\varphi}}_{5,N,j}} + {\left( {L_{1,N,j} + L_{{5a},N,j} + L_{{5b},N,j}} \right){\overset{.}{\varphi}}_{6,N,j}} + {L_{1,N,j}{\overset{.}{\varphi}}_{2,N,j}} + {L_{1,N,j}{\overset{.}{\varphi}}_{3,N,j}} - {L_{{5a},N,j}{\overset{.}{\varphi}}_{8,{N - 1},j}}} = {\varphi_{5,N,j} - \varphi_{4,N,j} + {2\pi\; x_{e}a_{2,N,j}} - {L_{{5a},N,j}\left( {{\sum\limits_{k = 1}^{N}{\sin\;\varphi_{1,k,j}}} + {\sum\limits_{k = 1}^{N}{\sin\;\varphi_{2,k,j}}} - {\sum\limits_{k = 1}^{N - 1}{\sin\;\varphi_{4,k,j}}} - {\sum\limits_{k = 1}^{N}{\sin\;\varphi_{5,k,j}}}} \right)} + {\left( {L_{1,N,j} + L_{{5b},N,j}} \right)\sin\;\varphi_{5,N,j}} - {\left( {L_{1,N,j} + L_{{5\; a},N,j} + L_{{5b},N,j}} \right)i_{{c\; 6},N,j}\sin\;\varphi_{6,N,j}} - {L_{1,N,j}\sin\;\varphi_{2,N,j}} - {L_{1,N,j}i_{{c\; 3},N,j}\sin\;\varphi_{3,N,j}} + {L_{{5a},N,j}i_{{c\; 8},{N - 1},j}\sin\;\varphi_{8,{N - 1},j}}}} & {{Equation}\mspace{14mu}(65)}\end{matrix}$(L _(6a,N,j) +L _(6b,N,j)){dot over(φ)}_(6,N,j)=φ_(4,N,j)−φ_(5,N,j)−φ_(6,N,j)−(L _(6a,N,j) +L _(6b,N,j))i_(c3,N,j) sin φ_(3,N,j)  Equation (66)

To model the top right-hand corner cell 40 of array 70, cell 40 _(j,i)for j=1 and i=N (6 equations):

$\begin{matrix}{{{L_{{2a},N,1}{\overset{.}{\varphi}}_{1,N,1}} + {\left( {L_{1,N,1} + L_{{2\; b},N,1}} \right)\left( {{\sum\limits_{k = 1}^{N}{\overset{.}{\varphi}}_{1,k,1}} + {\sum\limits_{k = 1}^{N - 1}{\overset{.}{\varphi}}_{2,k,1}}} \right)} - {\left( {L_{1,N,1} + L_{{2a},N,1} + L_{{2b},N,1}} \right){\overset{.}{\varphi}}_{3,N,1}} + {L_{1,N,1}{\overset{.}{\varphi}}_{5,N,1}} - {L_{1,N,1}{\overset{.}{\varphi}}_{6,N,1}} + {L_{{2a},N,1}{\overset{.}{\varphi}}_{7,{N - 1},1}}} = {{N \times \left( {L_{1,N,1} + L_{{2b},N,1}} \right)i_{b}} + \varphi_{2,N,1} - \varphi_{1,N,1} + {2\pi\; x_{e}a_{1,N,1}} - {L_{{2a},N,1}\sin\;\varphi_{1,N,1}} - {\left( {L_{1,N,1} + L_{{2b},N,1}} \right)\left( {{\sum\limits_{k = 1}^{N}{\sin\;\varphi_{1,k,1}}} + {\sum\limits_{k = 1}^{N - 1}{\sin\;\varphi_{2,k,1}}}} \right)} - {L_{1,N,1}\sin\;\varphi_{5,N,1}} + {\left( {L_{1,N,1} + L_{{2a},N,1} + L_{{2b},N,1}} \right)i_{{c\; 3},N,1}\sin\;\varphi_{3,N,1}} + {L_{1,N,1}i_{{c\; 6},N,1}\sin\;\varphi_{6,N,1}} - {L_{{2\; a},N,1}i_{{c\; 7},{N - 1},1}\sin\;\varphi_{7,{N - 1},1}}}} & {{Equation}\mspace{14mu}(67)}\end{matrix}$

$\begin{matrix}{{{- {L_{{2a},N,1}\left( {\sum\limits_{k = 1}^{N - 1}{{\overset{.}{\varphi}}_{1,k,1}{\sum\limits_{k = 1}^{N}{\overset{.}{\varphi}}_{2,k,1}}}} \right)}} - {\left( {L_{1,N,1} + L_{{2b},N,1}} \right){\overset{.}{\varphi}}_{2,N,1}} - {\left( {L_{1,N,1} + L_{{2a},N,1} + L_{{2b},N,1}} \right){\overset{.}{\varphi}}_{3,N,1}} + {L_{1,N,1}{\overset{.}{\varphi}}_{5,N,1}} - {L_{1,N,1}{\overset{.}{\varphi}}_{6,N,1}} + {L_{{2a},N,1}{\overset{.}{\varphi}}_{7,{N - 1},1}}} = {{{- N} \times L_{{2a},N,1}i_{b}} + \varphi_{2,N,1} - \varphi_{1,N,1} + {2\pi\; x_{e}a_{1,N,1}} + {L_{{2a},N,1}\left( {{\sum\limits_{k = 1}^{N - 1}{\sin\;\varphi_{1,k,1}}} + {\sum\limits_{k = 1}^{N}{\sin\;\varphi_{2,k,1}}}} \right)} + {\left( {L_{1,N,1} + L_{{2b},N,1}} \right)\sin\;\varphi_{2,N,1}} - {L_{1,N,1}\sin\;\varphi_{5,N,1}} + {\left( {L_{1,N,1} + L_{{2a},N,1} + L_{{2b},N,1}} \right)i_{{c\; 3},N,i}\sin\;\varphi_{3,N,1}} + {L_{1,N,1}i_{{c\; 6},N,1}\sin\;\varphi_{6,N,1}} - {L_{{2a},N,1}i_{{c\; 7},{N - 1},1}\sin\;\varphi_{7,{N - 1},1}}}} & {{Equation}\mspace{14mu}(68)}\end{matrix}$(L _(3a,N,1) +L _(3b,N,1)){dot over(φ)}_(3,N,1)=φ_(2,N,1)−φ_(1,N,1)−φ_(3,N,1)−(L _(3a,N,1) +L _(3b,N,1))i_(c3,N,1) sin φ_(3,N,1)  Equation (69)

$\begin{matrix}{{{L_{{5a},N,1}{\overset{.}{\varphi}}_{4,N,1}} - {\left( {L_{1,N,1} + L_{{5b},N,1}} \right)\left( {{\sum\limits_{k = 1}^{N}{\overset{.}{\varphi}}_{1,k,1}} + {\sum\limits_{k = 1}^{N}{\overset{.}{\varphi}}_{2,k,1}} - {\sum\limits_{k = 1}^{N}{\overset{.}{\varphi}}_{4,k,1}} - {\sum\limits_{k = 1}^{N - 1}{\overset{.}{\varphi}}_{5,k,1}}} \right)} + {\left( {L_{1,N,1} + L_{{5a},N,1} + L_{{5b},N,1}} \right){\overset{.}{\varphi}}_{6,N,1}} + {L_{1,N,1}{\overset{.}{\varphi}}_{2,N,1}} + {L_{1,N,1}{\overset{.}{\varphi}}_{3,N,1}} - {L_{{5a},N,1}{\overset{.}{\varphi}}_{8,{N - 1},1}}} = {\varphi_{5,N,1} - \varphi_{4,N,1} + {2\pi\; x_{e}a_{2,N,1}} - {L_{{5a},N,1}\sin\;\varphi_{4,N,1}} + {\left( {L_{1,N,1} + L_{{5b},N,1}} \right)\left( {{\sum\limits_{k = 1}^{N}{\sin\;\varphi_{1,k,1}}} + {\sum\limits_{k = 1}^{N}{\sin\;\varphi_{2,k,1}}} - {\sum\limits_{k = 1}^{N}{\sin\;\varphi_{4,k,1}}} - {\sum\limits_{k = 1}^{N - 1}{\sin\;\varphi_{5,k,1}}}} \right)} - {\left( {L_{1,N,1} + L_{{5a},N,1} + L_{{5b},N,1}} \right)i_{{c\; 6},N,1}\sin\;\varphi_{6,N,1}} - {L_{1,N,1}\sin\;\varphi_{2,N,1}} - {L_{1,N,1}i_{{c\; 3},N,1}\sin\;\varphi_{3,N,1}} + {L_{{5a},N,1}i_{{c\; 8},{N - 1},1}\sin\;\varphi_{8,{N - 1},1}}}} & {{Equation}\mspace{14mu}(70)}\end{matrix}$

$\begin{matrix}{{{L_{{5a},N,1}\left( {{\sum\limits_{k = 1}^{N}{\overset{.}{\varphi}}_{1,k,1}} + {\sum\limits_{k = 1}^{N}{\overset{.}{\varphi}}_{2,k,1}} - {\sum\limits_{k = 1}^{N - 1}{\overset{.}{\varphi}}_{4,k,1}} - {\sum\limits_{k = 1}^{N}{\overset{.}{\varphi}}_{5,k,1}}} \right)} - {\left( {L_{1,N,1} + L_{{5b},N,1}} \right){\overset{.}{\varphi}}_{5,N,1}} + {\left( {L_{1,N,1} + L_{{5a},N,1} + L_{{5b},N,1}} \right){\overset{.}{\varphi}}_{6,N,1}} + {L_{1,N,1}{\overset{.}{\varphi}}_{2,N,1}} + {L_{1,N,1}{\overset{.}{\varphi}}_{3,N,1}} - {L_{{5a},N,1}{\overset{.}{\varphi}}_{8,{N - 1},1}}} = {\varphi_{5,N,1} - \varphi_{4,N,1} + {2\pi\; x_{e}a_{2,N,1}} - {L_{{5a},N,1}\left( {{\sum\limits_{k = 1}^{N}{\sin\;\varphi_{1,k,1}}} + {\sum\limits_{k = 1}^{N}{\sin\;\varphi_{2,k,1}}} - {\sum\limits_{k = 1}^{N - 1}{\sin\;\varphi_{4,k,1}}} - {\sum\limits_{k = 1}^{N}{\sin\;\varphi_{5,k,1}}}} \right)} + {\left( {L_{1,N,1} + L_{{5b},N,1}} \right)\sin\;\varphi_{5,N,1}} - {\left( {L_{1,N,1} + L_{{5a},N,1} + L_{{5b},N,1}} \right)i_{{c\; 6},N,1}\sin\;\varphi_{6,N,1}} - {L_{1,N,1}\sin\;\varphi_{2,N,1}} - {L_{1,N,1}i_{{c\; 3},N,1}\sin\;\varphi_{3,N,1}} + {L_{{5a},N,1}i_{{c\; 8},{N - 1},1}\sin\;\varphi_{8,{N - 1},1}}}} & {{Equation}\mspace{14mu}(71)}\end{matrix}$(L _(6a,N,1) +L _(6b,N,1)){dot over(φ)}_(6,N,1)=φ_(4,N,1)−φ_(5,N,1)−φ_(6,N,1)−(L _(6a,N,1) +L _(6b,N,1))i_(c3,N,1) sin φ_(3,N,1)  Equation (72)

In similar fashion of the above, in order to model the bottom right-handcorner cell of 2D SQIF array 70, cell 40 _(j,i) for j=M and i=N (6equations):

$\begin{matrix}{{{L_{{2a},N,M}{\overset{.}{\varphi}}_{1,N,M}} - {\left( {L_{1,N,M} + L_{{2b},N,M}} \right)\left( {{\sum\limits_{k = 1}^{N}{\overset{.}{\varphi}}_{4,k,{M - 1}}} + {\sum\limits_{k = 1}^{N}{\overset{.}{\varphi}}_{5,k,{M - 1}}} - {\sum\limits_{k = 1}^{N}{\overset{.}{\varphi}}_{1,k,M}} - {\sum\limits_{k = 1}^{N - 1}{\overset{.}{\varphi}}_{2,k,M}}} \right)} - {\left( {L_{1,N,M} + L_{{2a},N,M} + L_{{2b},N,M}} \right){\overset{.}{\varphi}}_{3,N,M}} + {L_{1,N,M}{\overset{.}{\varphi}}_{5,N,M}} - {L_{1,N,M}{\overset{.}{\varphi}}_{6,N,M}} + {L_{{2a},N,M}{\overset{.}{\varphi}}_{7,{N - 1},M}}} = {\varphi_{2,N,M} - \varphi_{1,N,M} + {2\pi\; x_{e}a_{1,N,M}} + {\left( {L_{1,N,M} + L_{{2b},N,M}} \right)\left( {{\sum\limits_{k = 1}^{N}{\sin\;\varphi_{4,k,{M - 1}}}} + {\sum\limits_{k = 1}^{N}{\sin\;\varphi_{5,k,{M - 1}}}} - {\sum\limits_{k = 1}^{N}{\sin\;\varphi_{1,k,M}}} - {\sum\limits_{k = 1}^{N - 1}{\sin\;\varphi_{2,k,M}}}} \right)} - {L_{{2a},N,M}\sin\;\varphi_{1,N,M}} + {\left( {L_{1,N,M} + L_{{2a},N,M} + L_{{2b},N,M}} \right)i_{{c\; 3},N,M}\sin\;\varphi_{3,N,M}} - {L_{1,N,M}\sin\;\varphi_{5,N,M}} + {L_{1,N,M}i_{{c\; 6},N,M}\sin\;\varphi_{6,N,M}} - {L_{{2a},N,M}i_{{c\; 7},{N - 1},M}\sin\;\varphi_{7,{N - 1},M}}}} & {{Equation}\mspace{14mu}(73)}\end{matrix}$

$\begin{matrix}{{{L_{{2a},N,M}\left( {{\sum\limits_{k = 1}^{N}{\overset{.}{\varphi}}_{4,k,{M - 1}}} + {\sum\limits_{k = 1}^{N}{\overset{.}{\varphi}}_{5,k,{M - 1}}} - {\sum\limits_{k = 1}^{N - 1}{\overset{.}{\varphi}}_{1,k,M}} - {\sum\limits_{k = 1}^{N}{\overset{.}{\varphi}}_{2,k,M}}} \right)} - {\left( {L_{1,N,M} + L_{{2b},N,M}} \right){\overset{.}{\varphi}}_{2,N,M}} - {\left( {L_{1,N,M} + L_{{2a},N,M} + L_{{2b},N,M}} \right){\overset{.}{\varphi}}_{3,N,M}} + {L_{1,N,M}{\overset{.}{\varphi}}_{5,N,M}} - {L_{1,N,M}{\overset{.}{\varphi}}_{6,N,M}} + {L_{{2a},N,M}{\overset{.}{\varphi}}_{7,{N - 1},M}}} = {\varphi_{2,N,M} - \varphi_{1,N,M} + {2\pi\; x_{e}a_{1,N,M}} - {L_{{2a},N,M}\left( {{\sum\limits_{k = 1}^{N}{\sin\;\varphi_{4,k,{M - 1}}}} + {\sum\limits_{k = 1}^{N}{\sin\;\varphi_{5,k,{M - 1}}}} - {\sum\limits_{k = 1}^{N - 1}{\sin\;\varphi_{1,k,M}}} - {\sum\limits_{k = 1}^{N}{\sin\;\varphi_{2,k,M}}}} \right)} + {\left( {L_{1,N,M} + L_{{2b},N,M}} \right)\sin\;\varphi_{2,N,M}} + {\left( {L_{1,N,M} + L_{{2a},N,M} + L_{{2b},N,M}} \right)i_{{c\; 3},N,M}\sin\;\varphi_{3,N,M}} - {L_{1,N,M}\sin\;\varphi_{5,N,M}} + {L_{1,N,M}i_{{c\; 6},N,M}\sin\;\varphi_{6,\; N,M}} - {L_{{2a},N,M}i_{{c\; 7},{N - 1},M}\sin\;\varphi_{7,{N - 1},M}}}} & {{Equation}\mspace{14mu}(74)}\end{matrix}$(L _(3a,N,M) +L _(3b,N,M)){dot over(φ)}_(3,N,M)=φ_(2,N,M)−φ_(1,N,M)−φ_(3,N,M)−(L _(3a,N,M) +L _(3b,N,M))i_(c3,N,M) sin φ_(3,N,M)  Equation (75)

$\begin{matrix}{{{L_{{5a},N,M}{\overset{.}{\varphi}}_{4,N,M}} - {\left( {L_{1,N,M} + L_{{5b},N,M}} \right)\left( {{\sum\limits_{k = 1}^{N}{\overset{.}{\varphi}}_{1,k,M}} + {\sum\limits_{k = 1}^{N}{\overset{.}{\varphi}}_{2,k,M}} - {\sum\limits_{k = 1}^{N}{\overset{.}{\varphi}}_{4,k,M}} - {\sum\limits_{k = 1}^{N - 1}{\overset{.}{\varphi}}_{5,k,M}}} \right)} + {\left( {L_{1,N,M} + L_{{5a},N,M} + L_{{5b},N,M}} \right){\overset{.}{\varphi}}_{6,N,M}} + {L_{1,N,M}{\overset{.}{\varphi}}_{2,N,M}} + {L_{1,N,M}{\overset{.}{\varphi}}_{3,N,M}} - {L_{{5a},N,M}{\overset{.}{\varphi}}_{8,{N - 1},M}}} = {\varphi_{5,N,M} - \varphi_{4,N,M} + {2\pi\; x_{e}a_{2,N,M}} - {L_{{5a},N,M}\sin\;\varphi_{4,N,M}} + {\left( {L_{1,N,M} + L_{{5b},N,M}} \right)\left( {{\sum\limits_{k = 1}^{N}{\sin\;\varphi_{1,k,M}}} + {\sum\limits_{k = 1}^{N}{\sin\;\varphi_{2,k,M}}} - {\sum\limits_{k = 1}^{N}{\sin\;\varphi_{4,k,M}}} - {\sum\limits_{k = 1}^{N - 1}{\sin\;\varphi_{5,k,M}}}} \right)} - {\left( {L_{1,N,M} + L_{{5a},N,M} + L_{{5b},N,M}} \right)i_{{c\; 6},N,M}\sin\;\varphi_{6,N,M}} - {L_{1,N,M}\sin\;\varphi_{2,N,M}} - {L_{1,N,M}i_{{c\; 3},N,M}\sin\;\varphi_{3,N,M}} + {L_{{5a},N,M}i_{{c\; 8},{N - 1},M}\sin\;\varphi_{8,{N - 1},M}}}} & {{Equation}\mspace{14mu}(76)}\end{matrix}$

$\begin{matrix}{{{L_{{5a},\; N,M}\left( {{\sum\limits_{k = 1}^{N}{\overset{.}{\varphi}}_{1,k,M}} + {\sum\limits_{k = 1}^{N}{\overset{.}{\varphi}}_{2,k,M}} - {\sum\limits_{k = 1}^{N - 1}{\overset{.}{\varphi}}_{4,k,M}} - {\sum\limits_{k = 1}^{N}{\overset{.}{\varphi}}_{5,k,M}}} \right)} - {\left( {L_{1,N,M} + L_{{5b},N,M}} \right){\overset{.}{\varphi}}_{5,N,M}} + {\left( {L_{1,N,M} + L_{{5a},N,M} + L_{{5b},N,M}} \right){\overset{.}{\varphi}}_{6,N,M}} + {L_{1,N,M}{\overset{.}{\varphi}}_{2,N,M}} + {L_{1,N,M}{\overset{.}{\varphi}}_{3,N,M}} - {L_{{5a},N,M}{\overset{.}{\varphi}}_{8,{N - 1},M}}} = {\varphi_{5,N,M} - \varphi_{4,N,M} + {2\pi\; x_{e}a_{2,N,M}} - {L_{{5a},N,M}\left( {{\sum\limits_{k = 1}^{N}{\sin\;\varphi_{1,k,M}}} + {\sum\limits_{k = 1}^{N}{\sin\;\varphi_{2,k,M}}} - {\sum\limits_{k = 1}^{N - 1}{\sin\;\varphi_{4,k,M}}} - {\sum\limits_{k = 1}^{N}{\sin\;\varphi_{5,k,M}}}} \right)} + {\left( {L_{1,N,M} + L_{{5b},N,M}} \right)\sin\;\varphi_{5,N,M}} - {\left( {L_{1,N,M} + L_{{5a},N,M} + L_{{5b},N,M}} \right)i_{{c\; 6},N,M}\sin\;\varphi_{6,N,M}} - {L_{1,N,M}\sin\;\varphi_{2,N,M}} - {L_{1,N,M}i_{{c\; 3},N,M}\sin\;\varphi_{3,N,M}} + {L_{{5a},N,M}i_{{c\; 8},{N - 1},M}\sin\;\varphi_{8,{N - 1},M}}}} & {{Equation}\mspace{14mu}(77)}\end{matrix}$(L _(6a,N,M) +L _(6b,N,M)){dot over(φ)}_(6,N,M)=φ_(4,N,M)−φ_(5,N,M)−φ_(6,N,M)−(L _(6a,N,M) +L _(6b,N,M))i_(c3,N,M) sin φ_(3,N,M)  Equation (78)

What is claimed is:
 1. A two-dimensional Superconducting QuantumInterference Filter (SQIF) array comprising: at least three cellsarranged in two dimensions; each of said cells having at least twobi-Superconducting Quantum Interference Devices (bi-SQUIDs); said atleast two bi-SQUIDs being merged together so that said bi-SQUIDs shareat least two bi-SQUID junctions and at least one inductance, whereinsaid at least one inductance is varied to maintain an over-all anti-peakresponse by adjusting a hole size of a ground plane under saidinductance.
 2. The SQIF array of claim 1 wherein said cells arediamond-shaped or hexagonal-shaped when viewed in top plan.
 3. The SQIFarray of claim 1 wherein each of said cells has at least four celljunctions, and further wherein each of said cells shares at least threecell junctions with an adjacent cell.
 4. The SQIF array of claim 1wherein each said bi-SQUID has a loop size, wherein said loop sizes areuniform, and wherein said bi-SQUIDs have non-uniform inductances, andfurther wherein said non-uniform inductances are modeled to maintainsaid overall anti-peak response for said SQIF array that is linear.
 5. Amethod for establishing a two-dimensional Superconducting QuantumInterference Filter (SQIF) array, comprising the steps of: A) providinga plurality of bi-Superconducting Quantum Interference Devices(bi-SQUIDs); B) merging at least two bi-SQUIDs to establish a pluralityof array cells; said step B) being accomplished so that said at leasttwo bi-SQUIDs share at least two bi-SQUID junctions and at least oneinductance, and so that said step B) establishes at least four celljunctions, wherein said at least one inductance is varied to maintain anoverall anti-peak response by adjusting a hole size of a ground planeunder said inductance; and, C) connecting said array cells in atwo-dimensional manner so that each array cell shares at least threecell junctions with another of said array cells.
 6. The method of claim5, wherein said array cells are diamond shaped or hexagonal shaped whenviewed in top plan.
 7. The method of claim 5, wherein each said bi-SQUIDhas a loop size, wherein said loop sizes are uniform, and wherein saidstep B) is accomplished using bi-SQUIDs having non-uniform inductances,and further comprising the step of: D) modeling said non-uniforminductances to maintain said overall anti-peak response for said SQIFarray that is linear.
 8. The method of claim 7, wherein said non-uniforminductances have a distribution σ of at least 30 percent.